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The Columbia University Libraries reserve the right to refuse to accept a copying order If, in Its judgement, fulfillment of the order would Involve violation of the copyright law. Author: Jevons, William Stanley Title: The principles of science Place: London Date: 1920 ^^■^xsa-g' MASTER NEGATIVE # COLUMBIA UNIVERSITY LIBRARIES PRESERVATION DIVISION BIBLIOGRAPHIC MICROFORM TARGET ORIGINAL MATERIAL AS FILMED - EXISTING BIBLIOGRAPHIC RECORD J53 Jevons, William Stanley, 1835-1882. The principles of science ; a treatise on logic and scien- tific method, by W. Stanlej'^ Jevons ... London, Macmil- lan and co., limited ; New York, The Macmillan co., 1900. xliv, 786 p. incl. front. 19i'-. 1. Logic. 2. Science — Methodology. i. Title. 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LSf.i #■' • "- g^^Wi^fT ' . ..A- •^•■ «4vr. V"*'-" *V' it'/* ft- . jf *■ =-"_,* 't'^i .^^4 3^# r^ ' ..■^■•V.'?|.* • '." f.-" * * * "T 'f > ,- • '. - ,*«^,% «^ft» t* *^.- ' ««44i?^fe*iSfe - Columbia ^Hnibersiitp c^.y LIBRARY School of Business THE PRINCIPLES OF SCIENCE. MACMILLAN AND CO., Limited LONDON . BOMBAY . CALCUTTA MELBOURNE THE MACMILLAN COMPANY NEW YORK . BOSTON . CHICAGO DALLAS . SAN FRANCISCO THE MACMILLAN CO. OF CANADA, LTa TORONTO i| '\ THE PRINCIPLES OF SCIENCE A TREATISE ON LOGIC AND SCIENTIFIC METHOD r tr-i«™» «» « I>1I4HHiK*II*1- i^Hii^'irA^Mii -ir m. t'W>% ■IBBf'»l«* iir* TUX UOQICM. UACUIMK. BY W. STANLEY JEVONS LL.D. (IDINB.), M.A. (lOND.)> F.R.8. )l ii MACMILLAN AND CO., LIMITED ST. MARTIN'S STREET, LONDON 1920 iiy < I COPTRIQHT. Firtt Edition (2 voU. %vo\ 1874. Second Bdition (I vol. erown Svo), 1S77. Urorinted wOA eorreetUmM, 1879, 1888, 1887, 1892, 190a JUpHnttd (8»o), 1906, 1907, 1918, 1920. T5^ C^rw. \ PBEFACE TO THE FIRST EDITION. It may be truly asserted that the rapid progress of the physical sciences during the last three centuries has not been accompanied by a corresponding advance in the theory of reasoning. Physicists speak familiarly of Scientific Method, but they could not readily describe what they mean by that expression. Profoundly engaged in the study of particular classes of natural phenomena, they are usually too much engrossed in the immense and ever-accumulating details of their special sciences to generalise upon the methods of reasoning which they unconsciously employ. Yet few will deny that these methods of reasoning ought to be studied, especially by those who endeavour to introduce scientific order into less successful and methodical branches of knowledge. The application of Scientific Method cannot be re- stricted to the sphere of lifeless objects. We must sooner or later have strict sciences of those mental and social phenomena, which, if comparison be possible, are of more interest to us than purely material phenomena. But it is the proper course of reasoning to proceed from the known to the unknown — from the evident to the obscure —from the material and palpable to the subtle and refined. The physical sciences may therefore be properly « I I 1 1 6 viii PREFACE TO THE FIRST EDITION. J o made the practice-ground of tlie reasoning powers, because they furnish us with a great body of precise and successful investigations. In these sciences we meet with happy instances of unquestionable deductive reasoning, of ex- tensive generalisation, of happy prediction, of satisfactory verification, of nice calculation of probabilities. We can note how the slightest analogical clue has been followed up to a glorious discovery, how a rash generalisation has at length been exposed, or a conclusive cxpcrimcntum crucis has decided the long-continued strife between two rival theories. In following out my design of detecting the general methods of inductive investigation, I have found that the more elaborate and interesting processes of quantitative induction have their necessary foundation in the simpler / science of Formal Logic. The earlier, and probably by^ Mar the least attractive part of this work, consists, there- fore, in a statement of the so-called Fundamental Laws of Thought, and of the all-important Principle of Substi- tution, of which, as I think, all reasoning is a develop- ment. The whole procedure of inductive inquiry, in its most complex cases, is foreshadowed in the combinational view of Logic, which arises directly from these fundamental principles. Incidentally I have described the mechanical arrangements by which the use of the iraportant form called the Logical Alphabet, and the whole working of the combinational system of Formal Logic, may be ren- dered evident to the eye, and easy to the mind and hand. The study both of Formal Logic and of the Theory of Probabilities has led me to adopt the opinion that there is no such thing as a distinct method of induction as contrasted with deduction, but that induction is simply an inverse employment of deduction. Within the last century a reaction has been setting in against the purely empirical procedure of Francis Bacon, and physicists have I I PREFACE TO THE FIRST EDITION. IX learnt to advocate the use of hypotheses. I take the extreme view of holding that Francis Bacon, although he correctly insisted upon constant reference to experience, had no correct notions as to the logical method by which from particular facts we educe laws of nature. I endea- vour to show that hypothetical anticipation of nature is an essential part of inductive inquiry, and that it is the Newtonian method of deductive reasoning combined with elaborate experimental verification, which has led to all the great triumphs of scientific research. In attempting to give an explanation of this view of Scientific Method, I have first to show that the sciences of number and quantity repose upon and spring from the simpler and more general science of Logic. The Theory of Probability, which enables us to estimate and calculate quantities of knowledge, is then described, and especial 1 attention is drawn to the Inverse Method of Probabilities, \ which involves, as I conceive, the true principle of in- / ductive procedure. No inductive conclusions are more than probable, and I adopt the opinion that the theory of probability is an essential part of logical method, so that the logical value of every inductive result must be deter- mined consciously or unconsciously, according to the principles of the inverse method of probability. The phenomena of nature are commonly manifested in quantities of time, space, force, energy, &c., and the observation, measurement, and analysis of the various quantitative conditions or results involved, even in a simple experiment, demand much employment of system- atic procedure. I devote a book, therefore, to a simple and general description of the devices by which exact measurement is effected, errors eliminated, a probable mean result attained, and the probable error of that mean ascertained. I then proceed to the principal, and probably the most interesting, subject of the book, illustrating successively the conditions and precautions requisite for f<:3 ^ i 7> •J I I / / I \ ' PREFACE TO THE FIRST EDITION. accurate observation, for successful experiment, and for the sure detection of the quantitative laws of nature. As it is impossible to comprehend aright the value of quantitative laws without constantly bearing in mind the \ degree of quantitative approximation to the truth probably s attained, I have devoted a special chapter to tlie Theory S)f Approximation, and however imperfectly I may have treated this subject, I must look upon it as a very essential Dart of a work on Scientific Method. It then remains to illustrate the sound use of hypo- thesis, to distinguish between the portions of knowledge wliich we owe to empirical observation, to accidental dis- covery, or to scientific prediction. Interesting questions arise concerning the accordance of quantitative theories and experiments, and I point out how the successive 'verification of an hypothesis by distinct methods of ex- periment yields conclusions approximating to but never attaining certainty. Additional illustrations of the geneml procedure of inductive investigations are given in a chapter on the Character of the Experimentalist, in which I endeavour to show, moreover, that the inverse use of deduction was really the logical method of such gi-eat masters of experimental inquiry as Newton, Huyghens, and Faraday. In treating Generalisation and Analogy, I consider the precautions requisite in inferring from one case to another, / or from one part of the universe to another part ; the ( validity of all such inferences resting ultimately upon \^he inverse method of probabilities. The treatment of Exceptional Phenomena appeared to afford an interesting subject for a further chapter illustrating the various modes in which an outstanding fact may eventually be explained. The formal part of the book closes with the subject of Classification, which is, however, very inadequately treated. I have, in fact, almost restricted myself to showing that all classification is fundamentally carried out upon the PREFACE TO THE FIRST EDITION. xi principles of Formal Logic and the Logical Alphabet described at the outset. In certain concluding remarks I have expressed the conviction which the study of Logic has by degrees forced upon my mind, that serious misconceptions are enteiiained by some scientific men as to the logical value of our knowledge of nature. We have heard much of what has been aptly called the Eeign of Law, and the necessity and uniformity of natural forces has been not uncommonly interpreted as involving the non-existence of an intelligent and benevolent Power, capable of inter- fering with th'iB course of natural events. Fears have been expressed that the progress of Scientific Method must therefore result in dissipating the fondest beliefs of the human heart. Even the 'Utility of lleligion' is seriously proposed as a subject of discussion. It seemed to be not out of place in a work on Scientific Method to allude to the ultimate results and limits of that method. I fear that I have very imperfectly succeeded in expressing my strong conviction that before a rigorous logical scrutiny the Keign of Law will prove to be an unverified hypo- thesis, the Uniformity of Nature an ambiguous expression, the certainty of our scientific inferences to a great extent a delusion. The value of science is of course very high, while the conclusions are kept well within the limits of the data on which they are founded, but it is pointed out that our experience is of the most limited character com- pared with what there is to learn, while our mental powers seem to fall infinitely short of the task of comprehending and explaining fully the nature of any one object. I draw the conclusion that we must interpret the results/ of Scientific Method in an afifirmative sense only. Ours] must be a truly positive philosophy, not that false nega-/ tive philosophy which, building on a few material facts,^ presumes to assert that it has compassed the bounds of existence, while it nevertheless ignores the most/ ]( ill xfi PREFACE TO THE FIRST EDITION unquestionable phenomena of tlie human mind and feel- ings. It is approximately certain that in freely employing illustrations drawn from many different sciences, I have frequently fallen into errors of detail. In this respect I must throw myself upon the indulgence of the reader, who will bear in mind, as I hope, that the scientific fact^ are generally mentioned purely for the purpose of illus- tration, so that inaccuracies of detail will not in the majority -^f cases affect the truth of the general principles illustrated. ti It fii J) r December i«;, i87». M 1 PREFACE TO TUE SECOND EDITION. V Few alterations of importance have been made in pre- paring this second edition. Nevertheless, advantage has bcxjn taken of the opportunity to revise very carefully both the language and the matter of the book. Cor- respondents and critics having pointed cut inaccuracies of more or leis importance in the first edition, suitable corrections and emendations have been made. I am under obligations to Mr. C. J. Monro, M.A., of Bamet, and to Mr. W. H. Brewer, MA., one of Her Majesty's Inspectors of Schools, for numerous corrections. Among several additions which have been made to the text, I may mention the abstract (p. 143) of Professor Clifford's remarkable investigation into the number of types of compound statement involving four classes of objects. This inquiry carries forward the inverse logical problem described in the preceding sections. Again, the need of some better logical method than the old Barbara Celarent, &c., is strikingly shown by Mr. Venn's logical problem, described at p. 90. A great number of candidates in logic and philosophy were tested by Mr. Venn with this problem, which, though simple in reality, was solved by very few of those who were ignorant of Boole's Logia Other evidence could be adduced by Mr. Venn of the need for some better means of logical training. To enable the Ml f I j XIT PREFACE TO THE SECOND EDITION. logical student to test his skUl in the solution of inductive logical problems, I have given (p. 127) a series of ten problems graduated in difficulty. To prevent misapprehension, it should be mentioned that, throughout this edition, I have substituted the name Zoffical Alphabet for Zor/ical Ahecedarium, the name applied in the first edition to the exhaustive series of logical combinations represented in terms of A, B, C, D (p. 94). It was objected by some readers that Ahecedarium \a a bug and unfamiliar name. To the chapter on Units and Standards of Measure- ment, I have added two sections, one (p. 325) containing a bnef statement of the Theory of Dimensions, and the other (p. 319) discussing Professor Clerk Max weU's very original suggestion of a Natural System of Standards for the measurement of space and time, depending upon the length and rapidity of waves of light. In my description of the Logical Machine in tho Philosophical Trarisactions (vol. 160, p. 498), I said— " It is rarely indeed that any invention is made without some anticipation being sooner or later discovered ; but up to the present time I am totally unaware of even a single previous attempt to devise or construct a macliiue which should perform the operations of logical inference ; and it IS only, I believe, in the satirical writings of Swift that an allusion to an actual reasoning machine is to be found." Before the paper was printed, however, I was able to refer (p. 518) to the ingenious designs of the late Mr. Alfred Smee as attempts to represent thought mechanically. Mr. Smee's machines indeed were never constructed, and, if constructed, would not have performed actual logical inference. It has now just come to light, however, that the celebrated Lord Stanhope actually did construct a mechanical device, capable of representing syUogistic mferences in a concrete form. It appears that logic was one of the favourite studies of this truly original and ingenious noblecmu. There remain fragments of a logical" M \H PREFACE TO THE SECOND EDITION. XV work, printed by the Earl at his own press, which show that he had arrived, before the year 1800, at the principle of the quantified predicate. He puts forward this prin- ciple in the most explicit manner, and proposes to employ it throughout his syllogistic system. Moreover, he con- verts negative propositions into affirmative ones, and represents these by means of the copula " is identic with." Thus he anticipated, probably by the force of his own unaided insight, the main points of the logical method originated in the works of George Bentham and George Boole, and developed in this work. Stanhope, indeed, has no claim to priority of discovery, because he seems never to liave published his logical writings, although they were put into print. There is no trace of them in the British Museum Library, nor in any other library or logical work, 80 far as I am aware. Both the papers and the logical contrivance have been placed by the present Earl Stanhope in the hands of the Kev. Eobert Harley, F.R.S., who will, I hope, soon publish a description of them.^ By the kindness of Mr. Harley, I have been able to examine Stanhope's logical contrivance, called by him the Demonstrator. It consists of a square piece of bay- wood with a square depression in the centre, across which two slides can be pushed, one being a piece of red glass, and the other consisting of wood coloured gray. The extent to which each of these slides is pushed in is indicated by scales and figures along the edges of the aperture, and the simple rule of inference adopted by Stanhope is : " To the gi-ay add the red and subtract the holon^' meaning by holon ipkov) the whole width of the aperture. This rule of inference is a curious anticipation of De Morgan's numerically definite syllogism (see below, p. 168), and of inferences founded on what Hamilton called " Ultra-total distribution." Another curious point about Stanhope's * Since tho above was written Mr. Harley has read an account of Stan- hope's logical remains at the Dublin Meeting (1878) of the British Association. The mper will be printed in Mind. (Note added November. l87».) 1 r- f \\\ I ) i tri PREFACE TO THE SECOND EDITION. device is, that one slide can be drawu out and pushed in again at right angles to the other, and the overlappino part of the slides then represents the probability of a conclusion, derived from two premises of which the pro- babilities are respectively represented by the projecting parte of the slides. Thus it appeal^ that Stanhope had studied the logic of probabHity as well as that of certainty here agam anticipating, however obscurely, the recent progress of logical science. It wiU be seen, however, that between Stanhope's Demonstrator and my Logical Machine there is no resemblance beyond the fact that they both perform logical inference. In the first edition I inserted a section (vol i p. 25), on Anticipations of the Principle of Substitution," and I have reprmted that section unchanged in this edition (p. 21). I remark therein that, " In such a subject as lo-ic It IS hardly possible to put forth any opinions which ha've not been m some degree previously entertained. The germ at least of every doctrine wiU be found in eariier wntmgs, and novelty must arise chiefly in the mode of harmonising and developing ideas." I point out as Professor T. M Lindsay had previously done, that Beneke had employed the name and principle of substitution, and that doctrines closely approximating to substitution were stoted by the Port Eoyal Ix>gicians more than 200 years I have not been at aU surprised to learn, however, that other logicians have more or less distinctly stated this principle of substitution during the last two centuries As my friend and successor at Owens CoUege, Professor Adamson, has discovered, this principle can be traced back to no less a philosopher than Leibnitz. The remarkable tract of Leibnitz,i entitled "Non inelegans Specmien Demonstrandi in Abstractis," commences at once with a defimtion corresponding to the principle :— i^tT "^^ ^^'"^"^ ^ ^^. Krdxuaan. Par. I. Ben,liBi, PREFACE TO THE SECOND EDITION XVII " Eadem sunt quorum unum potest substitui alteri salva veritate. Si sint A et B, et A ingrediatur aliquam pro-" positionem veram, et ibi in aliqiio loco ipsius A pro ipso substituendo B fiat nova propositio seque itidem vera, idque semper succedat in quacunque tali propositione, A et B dicuntur esse eadem ; et contra, si eadem sint A et B, procedet substifutio quam dixi." Leibnitz, then, explicitly adopts the principle of sub- stitution, but he puts it in the form of a definition, saying that those things are the same which can be substituted one for the other, without affecting the truth of the proposition. It is only after having thus tested the same- ness of things that we can turn round and say that A and B, being the same, may be substituted one for the other. It would seem as if we were here in a vicious circle ; for we are not aUowed to substitute A for B, unless we have ascertained by trial tliat the result is a true proposition. Tlie difficulty does not seem to be removed by Leibnitz' proviso, "idque semper succedat in quacunque tali pro- positione." How can we learn that because A and B may be mutually substituted in some propositions, they may therefore be substituted in others ; and what is the criterion of likeness of propositions expressed in the word " tali " ? Whether the principle of substitution is to be regarded as a postulate, an axiom, or a definition, is just one of tliose fun- damental questions which it seems impossible to settle in the present position of philosophy, but this uncertainty will not prevent our making a considerable step in logical science. Leibnitz proceeds to establish in the form of a theorem what is usually taken as an axiom, thus (Opera, p. 95) : •• Theorema 1. Quae sunt eadem uni tertio, eadem sunt inter se. Si A cc B et B ex: C, erit A ex C, Nam si in pi-opositione A cc B (vera ea hypothesi) substituitur C in locum B (quod facerc licet per Def. I. quia B oc C ex hypothesi) fiet A cc C. Q. E. Dcm." Thus Leibnitz precisely anticipates the mode of treating inference with two simple identities described at p. 5 1 of this work. b y <( 1 xvm PREFACE TO THE SECOND EDITION. PREFACE TO THE SECOND EDITION. XIX i i III j Even the mathematical axiom that 'equals added to equals make equals/ is deduced from the principle of substitution. At p. 95 of Erdmann's edition, we find : " Si eideni addantur coincidentia fiunt coincidentia. SiAoiB, erit A + C oz B ■\- 0. Nam si in propositione A ■{■ C cc A -f C (quae est vera per se) pro A semel substituas /? (quod facere licet per Def. I. quia A (x B) ^et A -\- G o: B •{ C Q. K Dem." This is unquestionably the mode of deducing the several axioms of mathematical reasoning from the higher axiom of substitution, which is explained in the section on mathematical inference (p. 162) in this work, and which had been previously stated in my StcbstittUion of Similars, p. 16. Tliere are one or two other brief tracts in which Leibnitz anticipates the modern views of logic Thus in the eighteenth tract in Erdmann's edition (p. 92), called "Fundamenta Calculi Ratiocinatoris, he says: "Inter ea quorum unum alteri substitui potest, sal vis calculi legibus, dicetur esse gequipollentiam." There is evidence, also, that he had arrived at the quantification of the predicate, and that he fully understood the reduction of the universal affirmative proposition to the form of an equation, which is the key to an improved view of logic. Thus, in the tract entitled "Difficultates Qujedam Logicae,"* he says : "Omne-<4 est ^; id est equivalent AB et A, sen A non B est non-ens." It is curious to find, too, that Leibnitz was fully ac- quainted with the Laws of Commutativeness and " Simpli- city " (as I have called the second law) attaching to logical symbols. In the * Addenda ad Specimen Calculi Univer- salis" we read as follows.* " Transpositio literarum in eodem termino nihil mutat, ut ah coincidet cum ha, sen animal rationale et rationale animal." ** Repetitio ejusdem litene in eodem termino est inutilis, ut h est aa; vel hh est a; homo est animal animal, vel homo homo est animaL Sufficit cnim dici a est h, seu homo est animal." Comparing this with what is stated in Boole's Mathe- matical Analysis of Logic, pp. 17-18, in his Laws of Thov^ht, p. 29, or in this work, pp. 32-35, we find that Leibnitz had arrived two centuries ago at a clear perception of the bases of logical notation. When Boole pointed out that, in logic, axe = a?, this seemed to mathematicians to be a paradox, or in any case a wholly new discovery; but here we have it plainly stated by Leibnitz. Tlie reader must not assume, however, that because Leibnitz correctly apprehended the fundamental principles of logic, he left nothing for modern logicians to do. On the contrary, Leibnitz obtained no useful results from his definition of substitution. When he proceeds to explain the syllogism, as in the paper on " Definitiones Logicae," ^ he gives up substitution altogether, and falls back upon the notion of inclusion of class in class, saying, " Inclu- dens includentis est includens inclusi, seu si A includit B ct B includit G, etiam A includet G." He proceeds to make out certain rules of the syllogism involving the distinction of subject and predicate, and in no important respect better than the old rules of the syllogism. Leibnitz* logical tracts are, in fact, little more than brief memoranda of investigations which seem never to have been followed out They remain as evidence of his wonderful sagacity, but it would be difficult to show that they have had any influence on the progress of logical science in recent times. I should like to explain how it happened that these logical writings of Leibnitz were unknown to me, until within the last twelve months. I am so slow a reader of Latin books, indeed, that my overlooking a few pages of Leibnitz' works would not have been in any case surprising. But the fact is that the copy of Leibnitz' works of which I made occasional use, was one of the edition of Dutens, contained in Owens College Library. The logical ti-acts in question were not printed in that * Erdmann, p. 102. • Ibid p. 98. * £rdjnann, p. loa b 2 ^-- ^Asi. XX PREFACE TO THE SECOND EDITION. PREFACE TO THE SECOND EDITION. 1X1 I I iti edition, and with one exception, they remained in manu- script in the Eoyal Library at Hanover, until edited by Erdmann, in 1839-40. The tract " DifiBcultates Queedam Logicse," though not known to Dutens, was published by Kaspe in 1765, in his collection called (Euvres PhUo- sophiques tie feu M^' Leibnitz: but this work had not come to my notice, nor does the tract in question seem to contain any explicit statement of the principle of substitution. It is, I presume, the comparatively recent publication of Leibnitz' most remarkable logical tracts which explains the apparent ignorance of logicians as regards their con- tents and importance. The most learned logicians, such as Hamilton and Ueberweg, ignore Leibnitz* principle of substitution. In the Appendix to the fourth volume of Hamilton's Lectures on Meta^physics and Logic, is given an elaborate compendium of the views of logical writers concerning the ultimate basis of deductive reasoning. Leibnitz is briefly noticed on p. 319, but without any hint of substitution. He is here quoted as saying, " What are the same with the same third, are the same with each other ; that is, if ^ be the same with B^ and G be the same with B, it is necessary that A and C should also be the same with one another. For this principle flows immediately from the principle of contradiction, and is the ground and basis of all logic ; if that fail, there is no longer any way of reasoning with certainty." This view of the matter seems to be inconsistent with that which he adopted in his posthumous tract. Dr. Thomson, indeed, was acquainted with Leibnitz* tracts, and refers to them in his Outline of the Necessary Laws of Thought. He calls them valuable ; nevertheless, he seems to have missed the really valuable point ; for in making two brief quotations,^ he omits all mention of the principle of substitution. Ueberweg is probably considered the best authority > Fifth Edition, i860, p. 158. concerning the history of logic, and in his well-known System of Logic and History of Logical Doctrines^ he gives some account of the principle of substitution, especially as it is implicitly stated in the Port Eoyal Logic. But he omits all reference to Leibnitz in this connection, nor does he elsewhere, so far as I can tind, supply the omission. His English editor. Professor T. M. Lindsay, in referring to my Svhstitution of Similars, points out how I was antici- pated by Beneke ; but he also ignores Leibnitz. It is thus apparent that the most learned logicians, even when writing especially on the history of logic, displayed ignorance of Leibnitz' most valuable logical writings. It has been recently pointed out to me, however, that the Rev. Robert Harley did draw attention, at the Not- tingham Meeting of the Biitish Association, in 1866, to Leibnitz' anticipations of Boole's laws of logical notation,* and I am informed that Boole, about a year after the pub- lication of his Laws of Thought, was made acquainted with these anticipations by R. Leslie Ellis. There seems to have been at least one other German logician who discovered, or adopted, the principle of sub- stitution. Reusch, in his Systema Logicum, published in 1734, laboured to give a broader basis to the Dictum de Omni et Nullo. He argues, that " the whole business of ordinary reasoning is accomplished by the substitution of ideas in place of the subject or predicate of the funda- mental proposition. This some call the equation of thoughts." But, in the hands of Reusch, substitution does not seem to lead to simplicity, since it has to be carried on according to the rules of Equipollence, Reciprocation, Subordination, and Co-ordination.' Reusch is elsewhere spoken of * as the " celebrated Reusch " ; nevertheless, I have not been able to * Section 120. « See his "Remarks on Boole's Mathematical Analysis of Loric." fP^t of tM s^h Meeting of the British Association, Transactuyns of ik(. Sections, pp. 3—6. / "^ ' Hamilton's Lectures, vol. iv. p. ug. * IbicL p. 326. . I KXll PREFACE TO THE SECOND EDITION. PREFACE TO THE SECOND EDITION xxm II find a copy of his book in London, even in the British Museum Library; it is not mentioned in the printed catalogue of the Bodleian Libraiy; Messrs. Asher have failed to obtain it for me by advertisement in Germany ; and Professor Adamson has been equally unsuccessful. From the way in which the principle of substitution is mentioned by Keusch, it would seem likely that other logicians of the early part of the eighteenth century were acquainted with it ; but, if so, it is still more curious that recent historians of logical science have overlooked the doctrine. It is a strange and discouraging fact, that tnie views of logic should have been discovered and discussed from one to two centuries ago, and yet should have remained, like George Bentham's work in this century, without influ- ence.on the subsequent progress of the science. It may be regarded as certain that none of the discoverers of the quantification of the predicate, Bentham, Hamilton, Thomson, De Morgan, and Boole, were in any way assisted by the hints of the principle contained in previous writers. As to my own views of logic, they were originally moulded by a careful study of Boole's works, as fully stated in my first logical essay.^ As to the process of substitution, it was not learnt from any work on logic, but is simply the process of substitution perfectly familiar to mathematicians, and with which I necessarily became familiar in the course of my long-continued study of mathematics under the late Professor De Morgan. I find that the Theory of Number, which I explained in the eighth chapter of this work, is also partially anticipated in a single scholium of Leibnitz. He first gives as an axiom the now well-known law of Boole, as follows :— " Axioma L Si idem secum ipso sumatur, nihil consti- tuitur novum, sen ^ + ^ oc A." Then follows thia » Pure TA>gic or t)u Logic of QualUtj apart from Quantity; with Remarks onBooU'e System, and on the Helatian of Logic and AlalhmuUice London, 1864, p. 3- remarkable scholium : " Equidem iu numeris 4 + 4 facit 8, seu bini nummi binis additi faciunt quatuor nummos, sed tunc bini additi sunt alii a prioribus ; si iidem essent nihil novi prodiret et perinde esset ac si joco ex tribus ovis facere vellemus sex numerando, primum 3 ova, deinde uno sublato residua 2, ac denique uno rursus sublato residuum." Translated this would read as follows : — "Axiom I. If the same thing is taken together with itself, nothing new arises, or A -h A== A. " Scholium. In numbers, indeed, 4+4 makes 8, or two coins added to two coins make four coins, but then the two added are different from the former ones ; if they were the same nothing new would be produced, and it would be just as if we tried in joke to make six eggs out of three, by counting firstly the three eggs, then, one being removed, counting the remaining two, and lastly, one being again removed, counting the remaining egg." Compare the above with pp. 156 to 162 of the present work. M. Littrd has quite recently pointed out ^ what he thinks is an analogy between the system of formal logic, stated in the following pages, and the logical devices of the celebrated Itaymond Lully. Lully's method of invention was described in a great number of mediaeval books, but is best stated in hisArs Compendiosa Inveniendi Veritatem, seu Ars Magna et Major. This method consisted in placing various names of things in the sectors of concentric circles, so that when the circles were turned, every possible combination of the things was easily produced by mechani- cal means. It might, perhaps, be possible to discover in this method a vague and rude anticipation of combinational logic; but it is well known that the results of Lully's method were usually of a fanciful, if not absurd character. A much closer analogue of the Logical Alphabet is probably to be found in the Logical Square, invented by » La Philcsovhic Positive Mai-Juin, 1877, torn, xviii. p. 456. I xxiv PREFACE TO THE SECOND EDITION. PREFACE TO THE SECOND EDITION. XIV |i 'M II } • John Christian I^nge, and described in a rare and un- noticed work by liira which I have recently fonnd in the British Museum.i This squai-e involved the principle of bifurcate classification, and was an improved form of the Ramean and Porphyrian tree (see below, p. 702). Lange seems, indeed, to have worked out his Logical Square into a mechanical form, and he suggests that it might l>e employed somewhat in the manner of Napier's Rones (p. 65). There is much analogy between his Square and my Abacus, but Lange had not arrived at a logical system enabling him to use his invention for logical inference in the maimer of the Logical Abacus. Another work of Lange is said to contain the first publication of the well known Eulerian diagrams of proposition and syllogism.' Since the first edition was published, an important work by Mr. George Lewes has appeared, namely, his Problems 0/ Life cnui Mind, which to a great extent treats of scientific method, and formulates the rules of philo- sophising. I should have liked to discuss the bearing of Mr. Lewes's views upon those here propounded, but I have felt it to be impossible in a book already filling nearly 800 pages, to enter upon the discussion of a yet more extensive book. For the same reason I have not been able to compare my own treatment of the subject of probability with the views expressed by Mr. Venn in his Logic of Chance. With Mr. J. J. Murphy's profound and remarkable works on Hahii and Intelligence, and on The Scientific Basis of Faith, I was unfortunately unac- ' quainted when I wrote the following pages. They can- not safely be overlooked by any one who wishes to comprehend the tendency of philosophy and scientific method in the present day. It seems desirable that I should endeavour to answer some of the critics who have pointed out what they Svl^*^'*'"'^ ^'^^"^ (?wa<fra« Logiei, &c., Gisste Haasorum, 1714, ' Sei Ueherxoeg'8 SysUm of Lo^e, &c., translated by Lindsay, p. 302. consider defects in the doctrines of this book, especially in tlie first part, which treats of deduction. Some of the notices of the work were indeed rather statements of its contents than critiques. Thus, I am much indebted to M. Louis Liard, Professor of Philosophy at Bordeaux, for the very careful exposition ^ of the substitutional view of logic which he gave in the excellent Revue Philosophique, edited by M. Ribot. (Mars, 1877, tom. iii. p. 277.) An equally careful account of the system was given by M. Riehl, Professor of Philos(^hy at Graz, in his article on "Die Englische Logik der Gegenwart," published in the Vierteljahrsschrift filr toissenschaftliche Philosophie. ( i Heft, Leipzig, 1876.) T should like to acknowledge also the careful and able manner in which my book was reviewed by the New York Daily Tribune and the New York Times. The most serious objections which have been brought against my treatment of logic have regard to my failure to enter into an analysis of the ultimate nature and origin of the Laws of Thought. The Spectator^ for instance, in the course of a careful review, says of the principle of substitution, " Surely it is a great omission not to discuss whence we get this great principle itself; whether it is a pure law of the mind, or only an approximate lesson of experience ; and if a pure product of the mind, whether there are any other products of the same kind, furnished by our knowing faculty itself." Professor Robertson, in his very acute review,^ likewise objects to the want of » Since the above wa.s written M. Liard has republislied this exposition M one chanter of an interesting and admirably lucid account of the progress of logical science m England. After a brief but clear introduc- IndA.t^'I i"^*'^ic%^T,'^^^l''^^'^^^ ^^"' ^°d others concerning Sirf « .V^""' w' ^'.f ^ describes m succession the logical systems of t^^r.^^r'^'^'l' "^?"^'«'V ^^ J^'"'-^"' ^^^1«' ^"'i that contained in the present work. The title of the book is as follows -.-Les Logideiui k^i".^ifT ^^r*""^:, »"^^^"^' ^^""^' Professeur de Philos^hie \, io7». (A ote added November, 1878.) -^xsctotor, September 19 1874, p. 1178. A second portion of the review api^eared m the same journal fur September 26, 1874, p. ,204 XXVI PREFACE TO THE SECOND EDITION. • PREFACE TO THE SECOND EDITION. xxvu H I psychological and jihilosophical analysis. ** If the book really corresponded to its title, Mr. Jevons could hardly have passed so lightly over the question, which he does not omit to raise, concerning those undoubted principles of knowledge commonly called the Laws of Thouglit .... Everywhere, indeed, he appears least at ease when he touches on questions properly pliilosophical ; nor is he satisfactory in his psychological references, as on pp. 4, 5, where he cannot commit himself to a statement without an accompaniment of 'probably,' 'almost,' or 'hardly.' Reservations are often very much in place, but there are fundamental questions on which it is proper to make up one's mind." These remarks appear to me to be well founded, and I must state why it is that I have ventured to publish an extensive work on logic, without properly making up my mind as to the fundamental- nature of the reasoniuf process. The fault after all is one of omission mther than of commission. It is open to me on a future occasion to supply the deficiency if I should ever feel able to under- take the task. P>ut I do not conceive it to be an essential part of any treatise to enter into an ultimate analysis of its subject matter. Analyses must always end somewhere. There were good treatises on light which described the laws of the phenomenon correctly before it was known whether light consisted of undulations or of projected particles. Now we have treatises on the Undulatory Tlieory which are very valuable and satisfactory, although they leave us in almost complete doubt as to what the vibrating medium really is. So I think that, in the present day, we need a correct and scientific exhibition of the formal laws of thought, and of the forms of reasoning based on them, although we may not be able to enter into any complete analysis of the nature of those laws. What would the science of geometry be like now if the Greek geometers had decided that it was improper to publish any propositions before they had decided on the nature of an axiom? Where would the science of aiithmetic be now if an analysis of the nature of number itself were a necessary preliminary to a development of the results of its laws ? In recent times there have been enormous additions to the mathematical sciences, but very few attempts at psychological analysis. In the Alex- andrian and early mediajval schools of philosophy, much attention was given to the nature of unity and plurality chiefly called forth by the question of the Trinity. In the last two centuries whole sciences have been created out of the notion of plurality, and yet speculation on the nature of plurality has dwindled away. This present treatise contains, in the eighth chapter, one of the few recent attempts to analyse the notion of number itself. If further illustration is needed, I may refer to the differential calculus. Nobody calls in question the formal truth of the results of that calculus. All the more exact and successful parts of physical science depend upon its use, and yet the mathematicians who have created so great a body of exact truths have never decided upon the basis of the calculus. What is the nature of a limit or the nature of an infinitesimal? Start the question among a knot of mathematicians, and it will be found that hardly two agree, unless it is in regarding the question itself as a trifling one. Some hold that there are no such things as infinitesimals, and that it is all a question of limits. Others would argue that the infinitesimal is the necessary outcome of the limit, but various shades of intermediate opinion spring up. Now it is just the same with logic. If the forms of deductive and inductive reasoning given in the earlier part of this treatise are correct, they constitute a definite addition ta logical science, and it would have been absurd to decline to publish such results because I could not at the same time decide in my own mind about the psy- chology and philosophy of the subject. It comes in short to tliis, that my book is a book on Formal Logic and I I .^ (' IH xxviii PREFACE TO THE SECOND EDITION. Scientific Method, and not a book on psychology and philosophy. It may be objected, indeed, as the Spectator objects, that Mill's System of Logic is particularly strong in the discussion of the psychological foundations of reasoning, so that Mill would appear to have successfully treated that which I feel myself to be incapable of attempting at present. If Mill's analysis of knowledge is correct, then I have nothing to say in excuse for my own deficiencies. But it is well to do one thing at a time, and therefore I have not occupied any considerable part of this book with controversy and refutation. What 1 have to say of Mill's logic will be said in a separate work, in which his analysis of knowledge will be somewhat minutely analysed. It will then be shown, I believe, that Mill's psychological and philosophical treatment of logic has not yielded such satisfactory results as some writers seem to believe.^ Various minor but still important criticisms were made by Professor Robertson, a few of which have been noticed in the text (pp. 27, loi). In other cases his objections hardly admit of any other answer than such as consists in asking the reader to judge between the work and the criticism. Thus Mr. Robertson asserts* that the most complex logical problems solved in this book (up to p. 102 of this edition) might be more easily and shortly dealt with upon the principles and with the recognised methods of the traditional logic. The burden of proof here Jies upon Mr. Robertson, and his only proof consists in a single case, where he is able, as it seems to me accidentally, to get a special conclusion by the old form of dilemma. It would be a long labour to test the old logic upon every result obtained by my notation, and I must leave such * Portions of this work, have already been pablished in my articles entitled "John Stuart Mill's Philosophy Tested," printed in the ConUm- porary lUview for December, 1877, vol. xxxi. p. 167, and for January anri Apnl, 1878, vol. xxxL p. 256, and vol. xxxii. p. 88. (Nolo added iu November, 1878.) « Mind, vol. L p. 222 PREFACE TO THE SECOND EDITION. XXIX readers as are well acquainted with the syllogistic logic to pronounce upon the comparative simplicity and power of the new and old systems. For other acute objections brought forward by Mr. Robertson, I must refer the reader to the article in question. One point in my last chat)ter, that on the Results and Limits of Scientific Method, has been criticised by Professor W. K. Clifford in his lecture 1 on " The First and the Last Catastrophe." In vol. ii. p. 438 of the first edition (p. 744 of this edition) I referred to certain inferences drawn by eminent physicists as to a limit to the antiquity of the present order of things. " According to Sir W. Thomson's deductions from Fourier's theory of heat, we can trace down the dissipation of heat by con- duction and radiation to an infinitely distant time when all things will be uniformly cold. But we cannot similarly trace the Heat-liistory of the Universe to an infinite distance in the past. For a certain negative value of tlie time, the formulso give impossible values, indicating that there was some initial distribution of heat which could not have resulted, according to known laws of nature, from any previous distribution." Now according to Professor Clifford I have here mis- stated Thomson's results. "It is not according to the known laws of nature, it is according to the known laws of conduction of heat, that Sir William Thomson is speak- ing. ... All these physical writers, knowing what they were writing about, simply drew such conclusions from the facts which were before them as could be reasonably drawn. They say, here is a state of things which could not have been produced by the circumstances we are at present investigating Then your speculator comes, he reads a sentence and says, ' Here is an opportunity for me to have my fling.' And he has his fling, and makes a purely baseless theory about the necessary origin of the nJ. ^'T^^'^d^tly Review New Series. April 1875. p. 480. Lecture ro- printed by the Sunday Lecture Society, p. 24. M ,1 J\ .!^' XXX PREFACE TO THE SECOND EDITION. present order of nature at some definite point of timo, which might be calculated." Professor Clifford proceeds to explain that Thomson's formulfB only give a limit to the heat history of, say, the earth's cnist in the solid stat^. We are led back to the time when it became solidified from the fluid condition. There is discontinuity in the histoiy of the solid matter, but still discontinuity which is within our comprehension. Still further back we should come to discontinuity again, when the liquid was formed by the condensation of heated gaseous matter. Beyond that event, however, there is no need to suppose further discontinuity of law, for the gaseous matter might consist of molecules which had been falling together from different parts of space through infinite past time. As Professor Clifford says (p. 481) of the bodies of the universe, " What they have actually done is to fall together and get solid. If we shoiild reverse the process we should see them separating and getting cool, and as a limit to that, we should find that all these bodies would be resolved into melecules, and all these would be flying away from each other. There would be no limit to that process, and we could trace it as far back as ever we liked to trace it." Assuming that I have erred, I should like to point out that I have erred in the best company, or more strictly, being a speculator, I have been led into error by the best physical writers. Professor Tait, in his Sketch of Ther- modynamics, speaking of the laws discovered by Fourier for the motion of heat in a solid, says, " Their mathematical expressions point also to the fact that a uniform distribu- tion of heat, or a distribution tending to become uniform, must have arisen from some primitive distribution of heat of a kind not capable of being produced by known laws from any previous distribution." In the latter words it will be seen that there is no limitation to the laws of conduction, and, although I had carefully refeiTed to Sir W. Thomson's original paper, it is not unnatural PREFACE TO THE SECOND EDITION. xxxi that I should take Professor Tait's interpretion of its meaning.^ In his new work On some, Recent Advances in Physical Science^ Professor Tait has recurred to the subject as follows : * " A profound lesson may be learned from one of the earliest little papers of Sir W. Thomson, published while he was an undergraduate at Cambridge, where he shows that Fourier's magnificent treatment of the con- duction of heat [in a solid body] leads to formulae for its distribution which are intelligible (and of course capable of being fully verified by experiment) for all time future, but which, except in particular cases, when extended to time past, remain intelligible for a finite period only, and then indicate a state of things which could not have resulted under known laws from any conceivable previous distribution [of heat in the body]. So far as heat is concerned, modern investigations have shown that a previous distribution of the ma^/<jr involved may, by its potential energy, be capable of producing such a state of things at the moment of its aggregation ; but the example is now adduced not for its bearing on heat alone, but as a simple illustrntion of the fact that all portions of our Science, especially that beautiful one, the Dissipation of Energy, point unanimously to a beginning, to a state of things incapable of being derived by present laws [of tangible matter and its energy] from any conceivable previous arrangement." As this was published nearly a year after Professor CUfford's lecture, it may be infeiTed Txli!';^ Thomson's words are as follows {CamhHdge Mathematical ZrZl:, r- '^^,^'J°^- »»• ^' ,^74). "When x is ne^tivo, the state [f.L wM ? T''''^ ^. *^? '^""^^ ^^ Siny possible distribution of tempera- Ho^pn^« f \^ ^, P^^^^o'^-'ly existed." There is no limitation in the St« of ll f ?-'^' *?^ conduction, but, as the whole paper treats of the i^*T.V l?^°l 1- **''" o "" ",^^'^' '^ "'^y »° <^o"^t be understood that there ourn^fnr PM "• ^^S ^^° t ?"^°"^ l^^P^'- «« the subject in the same KllmiSr'^' '^ '''- "• ^- ''' ^''^'^ ^eain ^here is no ex- Tait'8^coSo^1•on.'^n^l^*"'"u'f ^' "'^ ^" ^^'^ original, and show Professor ti^tlf'a^^Tn oT^^^ ',68.r^'^" "'^''^ '' '^^ ''''''''''' ''''' «"^j-^ » If SXX1I PREFACE TO THE SECOND EDITION. */ that Professor Tait adheres to his original opinion that the theory of heat does give evidence of " a beginning." I may add that Professor Clerk Maxwell's words seem to countenance the same view, for he says,^ " This is only one of the cases in which a consideration of the dissi- pation of energy leads to the determination of a superior limit to the antiquity of the observed order of things." The expression " observed order of things " is open to much ambiguity, but in the absence of qualification I should take it to include the aggregate of tlie laws of nature known to us. I should interpret Professor Maxwell as meaning that the tlieory of heat indicates the occuiTence of some event of which our science cannot give any further explanation. The physical writers thus seem not to be so clear about the matter as Professor Clifford assumes. So far as I may venture to form an independent opinion on the subject, it is to the eflect that Professor Clifford is right, and that the known laws of nature do not enable us to assign a " beginning." Science leads us backwards into infinite past duration. But that Professor Clifford is right on this point, is no reason why we should suppose him to be right in his other opinions, some of which I am sure are wrong. Nor is it a reason why other parts of my last chapter should be wiong. The question only affects the single paragraph on pp. 744-5 of this book, which might, I believe, be struck out without necessitating any alteration in the rest of the text. It is always to be remembered that the failure of an argu- ment in favour of a proposition does not, generally speaking, add much, if any, probability to the contra- dictory proposition. I cannot conclude without expressing my acknowledgments to Professor Clifford for his kind expressions regarding my work as a whole. * Theory of Heat, 1871, p. 24$. 2, Th« Chestnuts, West Heath, Hamistead, N.W. August IS, 1877. CONTENTS- BOOK I. FORMAL LOGIC, DEDUCTIVE AND INDUCTIVE. CHAPTER L INTKODDOnON. SECTION p;^oE 1. Introduction j 2. The Powers of Miud concerned in the Creation of Science . . 4 8. Laws of Identity and Difference 5 4. The Nature of the Laws of Identity and Difference . . . .* 6 6. The Process of Inference 9 6. Deduction and Induction ] n 7. Symbolic Expression of Logical Inference 13 8. Kxpresaion of Identity and Dillerence 14 9. General Formula of Logical Inference .17 10. The Propagating Power of Similarity ! * *. 20 U. Anticipations of the Principle of Substitution ... ' . .* 21 12. The Logic of Kelatives . . ! ! 22 CHAPTER II. TUiMH. 1. Term* 2. Twofold mear.ini? of (Jenend Namen * ' «k 8. Abstract Terms .... ' ^ 4. Subauuti*; Terms . . \ [ '. \ \ . [ ' ' ' ^ 9 1 .1 1' I I I III XXXIT OONTBNTa il 1 1 ii i 8I0TI0N 5. Collective Tenus 6. Synthesis of Terms 7. Symbolic Expression of tbe Lnw of Contradiction 8. Certain Special Conditions of Logical Symbols . CHAPTER III. PAOK . 29 . 80 . 81 . 82 PROPOSmON'J 1. 2. 8. 4. 5. 6. Propositions . . . Simple Identities . Partial Identities . . Limited Identities . Negative rro]K)sitions Conversion of Propositions 86 .^7 40 42 43 46 47 7. Twofold Interpretation of Fropcutionw CHAPTER IV DRDUCrmS RKABONIKO. 1. Deductive Reasoning 49 2. Immediate Inference 50 8. Inference with Two Simple Identities 51 4. Inference with a Simple and a Paiaial Identity 53 5. Inference of a Partial from Two Partial Identities ... .55 6. On the Ellipsis of Terms in Partial Identities 57 7. Inference of a Simple from Two Partial Identities 58 8. Inference of a limited from Two Partial Identities .... 59 9. Miscellaneous Forms of Detluctive Inference .... . . CO *jO. Fallacies 62 CHAPTER V. DISJUNCTIVE PROrOSITIONS. y 1. Disjunctive Propositions 66 2. Expression of the Alternative Relation 67 8. Nature of the Alternative Relation 68 4. Laws of the Disjunctive Relation 71 5. Symbolic Expression of the Law of Duality 73 6. Various Forms of the Disjunctive Proposition 74 7. Inference by Disjunctive Propositioua 76 * OONTBNTa XXXV CHAPTER VL TBM INDIRECT METHOD OF INFERENCE. SECTIOlf PACK 1. The Indirect Method of Inference 81 2. Simple Illustrations 83 3. Employment of the Contrapositive Proposition 84 4. Contrapositive of a Simple Identity 86 5. Miscellaneous Examples of the Method 88 6. Mr. Venn's Problem 90 7. Abbreviation of the Process 91 8. The Logical Alphabet 94 9. The Logical Slate 95 10. Abstraction of Indifferent Circumstances 97 11. Illustrations of the Indirect Method 98 12. Second Example 99 13. Third Example 100 14. Fourth Example 101 15. Fifth Example 101 16. Fallacies Analysed by the Indirect Method 102 17. The logical Abacus 104 18. The L<^cal Machine 107 19. The Order of Premises 114 20. The Equivalence of Propositions ! . 115 21. The Nature of Inference 118 CHAPTER VIL IHDUCriON. 1. Induction 121 2. Induction an Inverse Operation 122 3. Inductive Problems for Solution by the Reader ! 126 4. Induction of Simple Identities ! .* 127 5. Induction of Partial Identities ' 130 6. Solution of the Inverse or Inductive Problem, involving Two Classes ^ ^ 134 7. The Inverse Logical Problem, involving Three Classes . . ] 137 8. Professor Clifford on the Types of Compound Statpaiar.t in- volving Four Classes 143 9. Distinction between Perfect and Imperfect Induction' .* .' .' 146 10. Transition from Perfect to Imperfect Induction 149 ' ll xxxn CONTJENT& OONTENTa ♦« mtTn ■aonON PAGE 6. Comparison of the Theory with Experience 206 6. Probable Deductive Arguments 209 7. Difficulties of the Theory 213 I / If I Ill It [} ■ (^ BOOK II. NUMBER, VARIETY, AND PROBABILITY. CHAPTER VIII. PUNOIPLKS OF NUMBXIL SECTION PAOI 1. Princn>Ie8 of Number 153 2. The Nature of Number ifig 8. Of Numerical Abstraction 158 4. Concrete and Abstract Number 159 5. Analogy of Logical and Numerical Terms 160 6. Principle of Mathematical Inference . . 162 7. Reasoning br Inequalities 165 8. Arithmetical Reasoning , 167 9. Numerically Definite Reasoning 168 10. Numerical meaning of Logical Conditioua 171 CHAPTER IX. TUB VARIETY OP NATUKX, OE THX DOOTBINK OP COMBINATIONS AHD PEBMUTAIIONB. 1. The Variety of Nature 178 2. Distinction of Combinations and Permutations .177 8. Calculation of Number of Combinations 180 4. The Arithmetical Triangle 182 5. Connexion between the Arithmetical Triangle and the logical Alphabet 189 C. Pctiidble Variety of Nature and Ark 190 7. Higher Orders of Variety 192 CHAPTER XI. PHILOSOPHY OF INDUCTIVE INFERENCE. 1. Philosophy of Inductive Inference ." . . . 218 2. Various Classes of Inductive Truths ... 219 8. The Relation of Cause and Effect 220 4. Fallacious Use of the Term Cause 221 5 Confusion of Two Questions , . 222 6. Definition of the Term Cause 224 7. Distinction of Inductive and Deductive Results 220 8. The Grounds of Inductive Inference 228 9. Illustrations of the Inductive Process 229 10. Geometrical Reasoning 283 11. Discrimination of Certainty and Probability ..!.!*. 235 CHAPTER XII. THE INDOCTIVE OR INVERSE APPLICATION OF THE THEORY OF PROBABILITy. 1. The Inductive or Inverse Application of the Theory . . 240 2. Pnnciple of the Inverse Method . . * 242 3. Simple Applications of the Inverse Method '. 244 i' m?® Theory of Probability in Astronomy. ...*.'*" 247 5. The General Inverse Problem " * 050 6. Simple Illustration of the Inverse Problem . . . [ ' 258 7. General Solution of the Inverse Problem. ...!*.** 255 8. Rules of the Inverse Method * * ' 257 9. Fortuitous Coincidences ......* 261 10. Summary of the Theory of Inductive Inference .*;.** *. 266 CHAPTER X. THEORY OF PROBABILITT. 1. Theory of Probability 2. Fundunental Principles of the Theory . . . 8. Rules for the Galenlatioii of ProbabUities . . 4. i'he Lutpcai Al|»habet in queetioas of Prvbability . 197 . 200 . 208 . S06 xxmil OONTIiWTa CX)NTENTS. xxjrtx f> \ I III }i BOOK ni. METHODS OF MEASUREMENT. CHAPTER Xlll. THE EXACT MEASUREMENT OF PHENOMENA. SECTION lAOB 1. The Exact Measurement of Phenomena . . . . ^ . . 270 2. Division of the Subject • 274 8. Continuous quantity ..../• 274 4. The Fallacious Indications of the Sensen .... . 276 5. Complexity of Quantitative Questions . 278 6. The Methods of Accurate Mftianromont .... . 282 7. Conditions of Accurate Measurement 282 8. Measuring Instruments .... 284 9. The Method of Repetition 288 10. Measurements by Natural Coincidence 292 11. Modes of Indirect Measuiei/ieAt 296 12. Comparative Use of Measuring Instruments 299 13. Systematic Performance of Meaaurementa 800 14. The Pendulum 302 15. Attainable Accuracy of Measurement 30S CHAPTER XIV. CHAPTER XV. ANALYSIS OF QTTANTITATITB PHENOMENA. SECTION P^GR 1. Analyaia of Quantitative Phenomena 335 2. lUuitrations of the Com^ication of Etfects 336 8. Methods of Eliminating Error 839 4. Method of Avoidance of Error .......... 340 6. Differential Method 844 6. Method of Correction 846 7. Method of Compensation 350 8. Method of Reversal 854 CHAPTER XVI. 1. 2. 3. 4. 5. 6. 7. THE METHOD OF MEANS. The Method of Means 357 Several Uses of the Mean Result 359 The Mean and the Average 3(j0 On the Average or Fictitious Mean ... ..... 368 The Precise Mean Result ' , . . 866 Determination of the Zero Point 363 Determination of Maximum Points . . . . . 8/1 I i 1. 2. 8. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 25. 16. 17. 18. 1» UNITS AND STANDARDS OF MEASUREMENT Units and Standards of Measurement ... 305 Stan.iard Unit of Time 807 The Unit of Space and the Bar Standard 312 The Terrestriiil Standard 814 The Pendulum Standard 315 Unit of Density 316 Unit of Mass .317 Natural System of Standards 319 Subsidiary Units 320 Derived Units 321 Provisional Units 823 Theory of Dimensions .... 825 Natural Constants ... 328 Mathematical Constants . , . . 880 Physical Constants ... ... 881 Astrunomioal Constants . ... 88S Terrestrial Numbers . ..... 888 Oiganic Numbers ... . •ijjS Social Numbers ... • • • "• CHAPTER XVII. THE LAW OF ERBOR. 1. The Law of Error 374 2. Establishment of the Law of Error . . . . . . . . 375 8. Herschel's Geometrical Proof ....*...** ' 377 4. lAplace's and Quetelet's Proof of the Law . . .* * * 378 o. Logical Origin of the Law of Error .... 383 6. Verification of the Law of Error . . 333 7. The Probable Mean Result .,...!...' i * 385 8. The Probable Error of Results ...*.*....!* 386 9. R«jeotion of the Mean Result . . ! ago 10. Method of Least Squares . . J ' lo y*!'^.^P***i*^^'^^''«'"y of Probability .;;*..'. .* 894 VL Detection of Constant Errors ....... 3M * rh "*i ! 1^ ii xl OONTBNTS. BOOK IV. INDUOTIVE INVESTIGATION. CHAPTER XVIII. 0B8ERVATI0K. SECTION P4QC 1. Observation . 899 2. Distinction of Observation and Experiment . . . • 400 3. Mental Conditions of Correct Observation 402 4. InstrumeDtal and Sensual Conditions of Correct Observation . 404 6. External Conditions of Correct Observation 407 6. Apparent Sequence of Events 409 7. Negative Arguments from Non^Obaervation 411 CHAPTER XIX. KrFKhlMKNT. • 1. Experiment 410 2. Exclusion of Indifferent CircumstanceA 419 3. Simplification of Experiments 422 4. Failure in the Simplification of Experiments 424 5. Removal of Usual Conditions 426 6. Interference of Unsuspected Conditions 428 7. Blind or Test Experiments 488 8. Negative Results of Experiment .... ... . 434 9. Limits of Experiment . 437 CHAPTER XX. UONTBKTS. xli CHAPTER XXI. imORT OF AFPROXIM ATIOK. 4SCTI0K 'AOK 1. Theory of Approximation ... 466 2. Substitution of Siinple hypotneses 458 3. Approximation to Exact Laws 462 4. Successive Approximations to Natural Conditions 465 5. Discovery of Hypothetically Simple Laws 470 6. Mathematical Principles of Approximation 471 7. Approximate Independence of Small Effects 476 8. Four Meanings of Equality 479 9. Arithmetic of Approximate Quantities 481 CHAPTER XXII <)UANTITAT[YS UiDUOTlON. 1. Quantitative Induction . ■ 483 2. F^bable Connexion of Varying Quantities .... . 484 3. Empirical Mathematical Laws 487 4. Discovery of Rational Formulae 489 6. The Graphical Method 492 6. Interpolation and Extrapolation 495 7. Illustrations of Empirical Quantitative I.Hwa 499 8. Simple Proportional Variation .... 601 CHAPTER XXIII. THS USE OF HTP0THE8IB. ,1 MBiHoo or rA&lAllOXIk 1. Method of Variations . . 489 2. The Variable and the Variant 440 3. ^leasurement of the Variable 441 1-. Maintenance of Similar Conditions 443 6. Collective Experiments 445 6. Periodic Variations 447 7. Combined Periodic Changes 450 8. Principle of Forced Vibrations 451 9. Integrated Variations . . . . , 468 1. The Use of Hypothesis , 504 2. Requisites of a good Hyjwthesis , 510 3. Possibility of Deductive Reasoning 511 4. Consistency with the Laws of Nature 614 6. Conformity with Facts ,,...! 516 6, Experimentum Crucis 1 *, ! 518 7. Descriptive Hypotheses ! ! * * 522 Illi OONTKNTS. CONTENTS. xliU ii / CHAPTER XXIV. . EMPIRICAL KirOWLIDOK, SXPLANAT10N AKD PRBDICTION. SKCnoX PAOl 1. Empirical Knowledge, Explanation and Prediction .... 525 2. Empirical Knowledge 526 8. Accidental Discovery 529 4. Empirical Observations subsequently Explained 532 5. Overlooked Results of Theory 534 6. Predicted Discoveries 536 7. Predictions in tlie Science of Light 588 8. Predictions from the Theory of Undulations 540 9. Prediction in other Sciences 542 10. Prediction by Inversion of Cause and Effect 545 11. Facts known only by Theory 547 CHAPTER XXV. ACCORDANCE OF QUANTITATIVE THEORIES. 1. Accordance of Quantitative Theories ... 551 2. Empirical Measurements 552 8. Quantities indicated by Theory, bat Empirically Measured . . 553 4. Explained Results of Measurwnent 554 5. Quantities determined by Theory and verified by Measurement 555 6. Quantities determined by Theory and not verified 556 7. Discordance of Theory and Experiment 558 8. Accordance of Measurements of Astronomical Distances . . 560 9. Selection of the best Mode of Measurement 563 10. Agreement of Distinct Modes of Measurement 564 11. Residual Phenomena 569 CHAPTER XXVI. CHARACTER OF THE EXPERIMENTALIST. 1. Character of the Experimentalist ...» 574 2. Error of the Baconian Method 576 3. Freedom of Theorising 577 4. The Newtonian Methcni, the True Organum 681 5. Candour and Courage of the Philosophic Mind 586 6. The Philosophic Character of Faraday 687 7. Reservation of Judgment .... ...... 592 BOOK V. GENERALISATION, ANALOGY. AND CLASSIFICATION. CHAPTER XXVII. GENS RALI8 AllON. SECTION l-AOE 1. Generalisation .... 594 2. Distinction of Generalisation and Analog\' . 596 3. Two Meanings of Generalisation 597 4. Value of Generalisation . . . 599 5. Comparative Generality of Properties 600 6. Uniform Properties of all Matter 603 7. Variable Properties of Matter 606 8. Extreme Instances of Properties 607 9. The Detection of Continuity 610 10. The Law of Continuity 615 11. Failure of the Law of Continuity 619 12. Negative Alignments on the Principle of Continuity .... 621 13. Tendency to Hasty Grcneralisation 623 CHAPTER XXVIII. ANA LOOT. 1. Analogy g27 2. Analogy as a Guide in Discoveiy 629 3. Analogy in the Mathematical Sciences . . . . . . . .' 631 4. Analogy in the Theory of Undulations 635 5. Analogy in Astronomy 533 6. Failures of Analogy 641 CHAPTER XXIX. EXCEPTIONAL PHENOMENA. 1. Exceptional Phenomena q^^ 2. Imaginary or False Exceptions .....!...* 647 3. Apparent but Congruent Exceptions . . ^ ! . G49 4. Singular Exceptions gro 5. Divergent Exceptions ] g^g 6. Accidental Exceptions '. . . . . ' 658 7. Novel and Unexplained Exceptions' ! . 661 8. Limiting Exceptions 553 9. Real Exceptions to Supposed Laws* . [ . aab 10. Unclassed Exceptions .W^. . i i .' ! 668 h Mhi CONTENTa » P| 1^ fnMi ■f >'' li !! fill- CHAPTER XXX. 0LA88IFI0ATI0N. SECTION PAOS 1. Classificatioii 673 2. Classification involving Induction . 675 3. Multiplicity of Modes of Classification . 677 4. Natural and Artificial Systems of Classification 679 5. Correlation of Properties 681 6. Classification in Crystallography . 685 7. Classification an Inverse and Tentative Operation 689 8. Symbolic Statement of the Theory of Classification .... 692 9. Bifurcate Classification 694 10. The Five Predicablcs 698 11. Summum Genus and Infima Species ... 701 12. The Tree of Porphyry 702 13. Does Abstractiou imply Generalisation ? 704 14. Discovery of Marks or CharacterisU'cs 708 15. Diagnostic Systems of Classification ... 710 16. Index Classifications 714 17. Classification in the Biological ScienoM 718 18. Classification by Types 722 19. Natural Genera and Species ... 724 20. Uui(^ue or Exceptional Objects 728 21. Limits of Classification 730 BOOK VI. CHAPTER XXXI. BEFLECTI0N8 ON THK RESULTS AND LIMITS OF 8CIENTIPI0 METHOD. Reflections on the Results and Limits of Scientific Method . . 735 The Meaning of Natural Law 787 Infiniteness of the Universe 788 The Indeterminate Problem of Creation 740 Hierarchy of Natural Laws 742 The Ambiguous Expression — *• Uniformity of Nature "... 745 Possible States of the Universe 749 S|)eculation8 on the Keconcentration of Energy 751 The Divergent Scope for New Discovery 752 Infinite Incompleteness of the Mathematical Sciences . . . 754 The Reign of Law in Mental and Social Phenomena . . . .759 The Theory of Evolution 761 Possibility of Divine Interference .... 765 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Conclnsion r66 ISTDEX 778 THE PEINCIPLES OF SCIENCE. CHAPTER I. INTRODUCTION. Science arises from the discovery of Identity amidst r\ Diversity. The process may be described in different ^ words, but our language must always imply the presence of one common and necessary element. In every act of inference or scientific method we ai-e engaged about a certain identity, sameness, similarity, likeness, resemblance, analogy, equivalence or equality apparent between two objects. It is doubtful whether an entirely isolated phenomenon could present itself to our notice, since there must always be some points of similarity between object and object. But in any case an isolated phenomenon could be studied to no useful purpose. The whole value of science consists in the power which it confers upon us of applying to one object the knowledge acquired from like objects ; and it is only so far, therefore, as we can discover and register resemblances that we can turn our observations to account. . -^^^^^ ^ * spectacle continually exhibited to our senses, in which phenomena are mingled in combinations of endless variety and novelty. Wonder fixes the mind's attention ; memory stores up a record of each distinct impression ; the powers of association bring forth the record when the like is felt again. By the higher faculties of juagment and reasoning the mind compares the new with i\ ^^ ^ f ! i ^ THE PRINCIPLES OF SCIENCE. [oflAP. the old, recognises essential identity, even when disguised by diverse circumstances, and expects to find again what was before experienced. It must be the ground of all reasoning and inference that what is true of one thing unit be IrvLe of its equivalent, and that under carefully ascertained conditions Nature repeats herself Were this indeed a Chaotic Universe, the powers of mind employed in science would be useless to us. Did Chance wholly take the place of order, and did all phenomena come out of an Infinite Lottery, to use Condorcet's ex- pression, there could be no reason to expect the like result in like circumstances. It is possible to conceive a world in which no two things should be associated more often, in the long run, than any other two things. The frequent conjunction of any two events would then be purely fortuitous, and if we expected conjunctions to recur con- tinually, we should be. disappointed. In such a world we might recognise the same kind of phenomenon as it ap- peared from time to time, just as we might recognise a marked ball as it was occasionally drawn and re-drawn from a ballot-box ; but the approach of any phenomenon would be in no way indicated by what had gone before, nor would it be a sign of what was to come after. In such a world knowledge would be no more than the memory of past coincidences, and the reasoning powers, if they existed at all, would give no clue to the nature of the present, and no presage of the future. Happily the Univei-se in which we dwell is not the result of chance, and where chance seems to work it is our own deficient faculties which prevent us from recog- nising the operation of Law and of Design. In the material framework of this world, substances and forces jiresent themselves in definite and stable combinations. Things are not in perpetual flux, as ancient philosophers hem. Element remains element; iron changes not into gold. With suitable precautions we can calculate upon finding the same thing again endowed with the same properties. The constituents of the globe, indeed, appear in almost endless combinations ; but each combination bears its fixed character, and when resolved is found to be the compound of definite substances. Misapprehensions must continually occur, owing to the limited extent of our exi)erieuce. We tj INTRODUCTION. can never have examined and registered possible exist- ences so thoroughly as to be sure that no new ones will occur and frustrate our calculations. The same outward appearances may cover any amount of hidden differences which we have not yet suspected. To the variety of sub- stances und powers diffused through nature at its creation, we should not suppose that our brief experience can assign a limit, and the necessary imperfection of our knowledge must be ever borne in mind. Yet there is much to give us confidence in Science. Tho wider our experience, the more minute our examination of the globe, the greater the accumulation of well-reasoned knowledge, — the fewer in all probability will be the failures of inference compared with the successes. Exceptions to the prevalence of Law are gradually reduced to Law themselves. Certain deep similarities have been detected among the objects around us, and have never yet been found wanting. As the means of examining distant parts of the universe have been acquired, .those similarities have been traced there as here. Other worlds and stellar systems may be almost incomprehensively different from ours in magnitude, condition and disposition of parts, and yet we detect there the same elements of which our own limbs are composed. The same natural laws can be detected in operation in every part of the universe within the scope of our instruments ; and doubtless these laws are obeyed irrespective of distance, time, and circumstance. It is the prerogative of Intellect to discover what is uni- form and unchanging in the phenomena around us. So far as object is different from object, knowledge is useless and inference impossible. But so far as object resembles object, we can pass from one to the other. In proportion as resemblance is deeper and more general, the com- manding powers of knowledge become more wonderful. Identity in one or other of its phases is thus always the bridge by which we pass in inference from case to case ; and it is my purpose in this treatise to trace out the various forms in which the one same process of reasoning presents itself in the ever-growing achievements of Scientific Method. B 2 THE PRINCIPLES OF SCIENCE. fOHAP. tO INTRODUCTION. in I The Powers of Mind concerrud in the OrecUion of Science, It is no part of the purpose of this work to investigate the nature of mind. People not uncommonly suppt»se that logic IS a branch of psychology, because reasoning is a mental operation. On the same ground, however, we might argue that all the sciences are branches of psy- chology. As will be further explained, I adopt the opinion of Mr. Herbert Spencer, that logic is really an objective science, like mathematics or mechanics. Only in an in- cidental manner, then, need I point out that the mental powers employed in the acquisition of knowledge are prob- ably three in number. They are substantially as Professor Bain has stated them i : — 1. The Power of Discrimination. 2. The Power of Detecting Identity. 3. The Power of Retention. We exert the first power in every act of perception. Hardly can we have a sensation or feeling unless we dis- criminate it from something else which preceded. Con- sciousness would almost seem to consist in the break between one state of mind and the next, just as an induced current of electricity arises fiom the beginning or the ending of the primary current. We are always engaged in cliscnmmation ; and the rudiment of thought which exists m the lower animals probably consists in their power of feeling difference and being agitated by it. Yet had we the power of discrimination only. Science could not be created. To know that one feeling differs from another gives purely negative information. It cannot teacli us what will happen. In such a state of intellect each sensation would stand out distinct from every other • there would be no tie, no bridge of affinity between them! We want a unifying power by which the present and the future may be linked to the past ; and this seems to be accomplished by a different power of mind. Lord Bacon has pointed out that different men possess in very different degrees the powers of discrimination and identification. It may be said indeed tliat discrimination necessarily implies the action of the opposite process of identification ; and so Tht Se7ws and the Intellect, Second Ed., pp. 5, 325, &c. it doubtless does in negative points. But there is a rare property of mmd which consists in penetrating the dis- guise of variety and seizing the common elements of sameness; and it is this property which furnishes the true measure of intellect. The name of" intellect " expresses the interlacing of the general and the single, which is the peculiar province of mind.i To cogitate is the I^tin co- agitate, restmg on a like metaphor. Logic, also is but another name for the same process, the peculiar work of reason ; for X0709 is derived from X^^^v, which like the L&tmlegere meant originally to gather. Plato said of this unifying power, that if he met the man who could detect t/u one %n the many, he would follow him as a god. Laws of Identity and Difference, At the base of all thought and science must lie the laws which express the very nature and conditions of the discriminating and identifying powers of mind These are the so-called Fundamental Laws of TJiought, usually stated as follows : — & > j 1. The Law of Identity. Whatever is, is. v/ 2. The Law of Contradiction. A thing cannot both be and not he. 3. The Law of Duality. A thing must eitUv he m^ not he. The first of these statements may perhaps be regarded as LnTZ>'''f 'i '^'"^^'^ ^^^^^' '^ '' fundamental a notion fectlt h' r ^''fiP-V^t "^."""^ ^^ ^°^ «^^^ent is peiv fectly Identical with itself, and, if any pei^on were unaware desorihT^r^ f ^^' r'^ " '^'""^^y'' ^^ ^^^1^ ^«t better ae^ribe it than by such an example. rJnofT'^-^^'^?''^''^?"^ ^^^^ contradictory attributes T^^.l'J'^ ^T^^ ^Sether. The same object may vary wVte . ^f r^^V^^'^'J ^'^'\'^ '''^y ^^ ^l^^k' and there White, at one time it may be hard and at another time ToLar6^^"o?'sfxtt^^ ^ Scimce 0/ Language, Second Series, A'i(^hne;n?n'cr of « ^f^ll /- °' .^^^- "; P' ^7- The view of the etymolo mTx m^y I intellect" is g.ren above on the authority of Professor to wh^ch h« lit''-";'/;- ^' 'PP^'"^ "^ *^^ ''"^'''^'y ^Pi»^on, accoSg <^<I l^wiln Y'a- ''¥^}^''' ?"^^n« to choose between, to s^e a ditfei? «ice between, to discrnnmate, instead of to unite. 11 M M ■!i THE PRINCIPLES OF SCIENCE. [chap. soft ; but at the same time and place an attribute cannot be both present and absent Aristotle truly described this law as the first of all axioms— one of which we need not seek for any demonstration. All truths cannot be proved, otherwise there would be an endless chain of demonstration ; and it is in self-evident truths like this that we find the simplest foundations. The third of these laws completes the other two. It asserts that at every step there are two possible alter- natives—presence or absence, affirmation or negation. Hence I propose to name this law the Law of Dualfty, for it gives to all the formulae of reasoning a dual character. It asserts also that between presence and absence, existence and non-existence, affirmation and negation, there is no third alternative. As Aristotle said, there can be no mean between opposite assertions: we must either affirm or deny. Hence the inconvenient name by which it has been known— The Law of Excluded Middle. It may be allowed that these laws are not three indepen- dent and distinct laws ; they rather express three different aspects of the same truth, and each law doubtless pre- supposes and implies the other two. But it has not hitherto been found possible to state these characters of identity and difference in less than the threefold formula. The reader may perhaps desire some information as to the mode in which these laws have been stated, or the way in which they have been regarded, by philosophers in different ages of the world. Abundant information on this and many other points of logical history will be found in Veherweg's Sf/stem of Logic, of which an excellent translation has been published by Professor T. M. Lindsay (see pp. 228-281). The Nature of the Laws of I<kntity and Difference, I must at least allude to the profoundly difficult ques- tion concerning the nature and authority of these Laws of Identity and Difference. Are they Laws of Thought or Laws of Things ? Do they belong to mind or to material nature ? On the one hand it may be said that science is a purely mental existence, and must therefore conform to the laws of that which formed it. Science is in the mind and INTRODUCTION. not in the things, and the properties of mind are therefore all important It is true that these laws are verified in the observation of the exterior world ; and it would seem that they might have been gathered and proved by general- isation, had they not already been in our possession. But on the other hand, it may well be urged that we cannot prove these laws by any process of reasoning or observation, because the laws themselves are presupposed, as Leibnitz acutely remarked, in the very notion of a proof. They are the prior conditions of all thought and all knowledge, and even to question their truth is to allow them true. Hartley ingeniously refined upon this argument, remarking that if the fundamental laws of logic be not certain, there must exist a logic of a second order whereby we may determine the degree of uncertainty : if the second logic be not certain, there must be a third ; and so on ad infinitum. Thus we must suppose either that absolutely certain laws of thought exist, or that there is no such thing as certainty whatever.^ Logicians, indeed, appear to me to have paid insufficient attention to the fact that mistakes in reasoning are always possible, and of not unfrequent occurrence. The Laws of Thought are often called necessary laws, that is, laws which cannot but be obeyed. Yet as a matter of fact, who is there that does not often fail to obey them ? They are the laws which the mind ought to obey rather than what it always does obey. Our thoughts cannot be the criterion of truth, for we often have to acknowledge mistakes in arguments of moderate complexity, and we sometimes only discover our mistakes by collision between our expectations and the events of objective nature. Mr. Herbert Spencer holds that the laws of logic are objective laws,* and he regards the mind as being in a state of constant education, each act of false reasoning or miscalculation leading to results which are likely to prevent siraUar mistakes from being again committed. I am quite inclined to accept such ingenious views ; but at the same time it is necessary to distinguish between the accumulation of knowledge, and the constitution of the '^^d which allows of the acquisition of knowledge. Before the mind can perceive or reason at all it must have * Hartley on Man, vol. i. p. 359, ' Frinciples of PsycJwlogy, Second Ed., vol. ii. p. 86. ' THE PRINCIPLES OF SCIENCE. [chap. I.] INTRODUCTION. 9 the conditions of thought impressed upon it. Before a mistake can be committed, the mind must clearly dis- tinguish the mistaken conclusion from all other assertions. Are not the Laws of Identity and Difference the prior conditions of all consciousness and all existence ? Must they not hold true, alike of things material and immaterial? and if so, can we say that they are only subjectively true or objectively true? I am inclined, in short, to regard them as true both " in the nature of thought and things/' as I expressed it in my first logical essay ; ^ and I hold that they belong to the common basis of all existence. But this is one of the most difficult questions of psychology and metaphysics which can be raised, and it is hardly one for the logician to decide. As the mathematician does not inquire into the nature of unity and plurality, but develops the formal laws of plurality, so the logician, as I conceive, must assume the truth of the Laws of Identity and Difference, and occupy himself in developing the variety of forms of reasoning in which their truth may be manifested. Again, I need hardly dwell upon the question whether logic treats of language, notions, or things. As reasonably might we debate whether a mathematician treats of symbols, quantities, or things. A mathematician certainly does treat of symbols, but only as the instruments whereby to facilitate his reasoning concerning quantities ; and as the axioms and riiles of mathematical science must be verified in concrete objects in order that the calcula- tions founded upon them may have any validity or utility, it follows that the ultimate objects of matliematical science are the things themselves. In like manner I conceive that the logician treats of language so far as it is essential for the embodiment and exhibition of thought. Even if reasoning can take place in the inner cx)nsciousness of man without the use of any signs, which is doubtful, at any rate it cannot become the subject of discussion until by some system of material signs it is manifested to other persons. The logician then uses words and symbols as instruments of reasoning, and leaves tlie nature and peculiarities of language to the grammarian. But signs again must * Pure Logic, or the Logic of Quality apart from Quantity, 1864, pp- 10, 16, 22, 29, 36, &C. correspond to the thoughts and things expressed, in order that they shall serve their intended purpose. We may therefore say that logic treats ultimately of thoughts and things, and immediately of the signs which stand for them. Signs, thoughts, and exterior objects may be regarded as parallel and analogous series of plienomena, and to treat any one of the three series is equivalent to treating either of the other series. 77ie Process of Inference. Xiie fundamental action of our reasoning faculties consists in inferring or carrying to a new instance of a phenomenon whatever we have previously known of its like, analogue, equivalent or equal. Sameness or identity presents itself in all degrees, and is known under various names; but the great rule of inference embraces all degrees, and affirms that so far as there exists sameness^ identity or likeness, what is true of one thing will he true^ of the other. The great difficulty doubtless consists inT ascertaining that there does exist a sufficient degree off likeness or sameness to warrant an intended inference;! and it will be our main task to investigate the conditions under which reasoning is valid. In this place I wish to point out that there is something common to all acts of inference, however different their apparent forms. The one same rule lends itself to the most diverse applications. The simplest possible case of inference, perhaps, occui-s in the use of a pattern, example, or, as it is commonly called, a sample. To prove the exact similarity of two portions of commodity, we need not bring one portion beside the other. It is sufficient that we take a sample which exactly represents the texture, appearance, and general nature of one portion, and according as this sample agrees or not with the other, so will the two portions of commodity agree or differ. Whatever is true as regards the colour, texture, density, material of the sample will be true of the goods themselves. In such cases likeness of quality is the condition of inference. Exactly the same mode of reasoning holds true of magnitude and figure. To compare the sizes of two objects, we need not lay them beside each other. A A v/ il 10 'HE rUlNCIPLES OF SCIENCE. [chap. staff, string, or other kind of measure may be employed to represent the length of one object, and according as it agrees or not with the other, so must the two objects agree or differ. In this case the proxy or sample represents length ; but the fact that lengths can be added and multiplied renders it unnecessary that the proxy should always be as large as the object. Any standard of convenient size, such as a common foot-rule, may be made the medium of comparison. The height of a church in one town may be carried to that in another, and objects existing immovably at opposite sides of the earth may be vicanously measured against each other. We obviously employ the axiom that whatever is true of a thing as regards its length, is true of its equal To every other simple phenomenon in nature the same principle of substitution is applicable. We may compare weights, densities, degrees of hardness, and degrees of all other qualities, in like manner. To ascertain whether two sounds are in unison we need not compare them directly, but a third sound may be the go-between. If a tunin<^- fork IS in unison with the middle C of York Minster organ, and we afterwards find it to be in unison with the same note of the organ in Westminster Abbey, then it follows that tl;e two organs are tuned in unison. The rule of inference now is, that what is tnie of the tunincr- fork as regards the tone or pitch of its sound, is true of any sound in unison with it. The skilful employment of this substitutive process enables us to make measurements beyond the powers of our senses. No one can count the vibrations, for instance, ^\i^A ^**^'^""PiP^- ^"^ we can construct an instrument called the siren, so that, while producing a sound of any pitch, it shall register the number of vibrations consti- tuting the sound. Adjusting the sound of the siren in unison with an organ-pipe, we measure indirectly the number of vibrations belonging to a sound of that pitch. To measure a sound of the same pitch is as good as to measure the sound itself. Sir David Brewster, in a somewhat similar manner, succeeded in measuring the refractive indices of irregular fragments of transparent minerals. It was a troublesome, and sometimes impracticable work to grind the minerals I.] INTRODUCTION. 11 into prisms, so that the power of refracting light could be directly observed ; but he fell upon the ingenious device of compounding a liquid possessing the same refractive power as the transparent fragment under examination. The moment when this equality was attained could be known by the fragments ceasing to reflect or refract light when immersed in the liquid, so that they became almost invisible in it. The refractive power of the liquid being then measured gave that of the solid. A more beautiful instance of representative measurement, depending im- mediately upon the principle of inference, could iiot be found.^ Throughout the various logical processes which we are about to consider— Deduction, Induction, Generalisation, Analogy, Classification, Quantitative Reasoning— we shall find the one same principle operating in a more or less disguised foruL Deduction and Indicction, The processes of inference always depend on the one same principle of substitution ; but they may nevertheless be distinguished according as the results are inductive or deductive. As generally stated, deduction consists in passing from more general to less general truths ; induc- tion is the contrary process from less to more genera, truths. We may however describe the difference in another manner. In deduction we are engaged in develop- V nig the consequences of a law. We learn the meaning; contents, results or inferences, which attach to any given proposition. Induction is the exactly inverse process. Uven certain results or consequences, we are required to discover the general law from which they flow. In a certain sense all knowledge is inductive. We can only learn the laws and relations of things in nature by observing those things. But the knowledge gained from the senses is knowledge only of particular facts, and we require some process of reasoning by which we may collBct out of the facts the laws obeyed by them. ' Brewster, Treaiiie on New , Philosophical InstrumenU, p. 27^ J^ncerning this method see also WheweU, Philosophy of the Inductive !^icn, vol. iL p. 355 ; Toinlinson, Philosophical Magazine, Fourth ^>«ne8, vol xl. p. 328 ; Tyndall, in Younians' Modem Culture, p. 16. f^ II THE PRINCIPLES OP SCfENCE. lii (OHAr. Experience gives us the materials of knowledge : induction digests those materials,, and yields us general knowledge. When we possess such knowledge, in the form of general propositions and natural laws, we can usefully apply the reverse process of deduction to ascertain the exact information required at any moment. In its ultimate loundation, then, all knowledge is inductive— in the sense that it is derived by a certain inductive reasoning from the facts of experience. It is nevertheless true, — and this is a point to which insufficient attention has been paid, that all reasoning IS founded on the principles of deduction. I call in question the existence of any method of reasoning which can be carried on without a knowledge of deductive pro- cesses. I shall endeavour to show that induction is really the inverse process of deduction. There is no mode of ascertaining the laws which are obeyed in certain pheno- mena, unless we have the power of determining what results would follow from a given law. Just as the process of division necessitates a prior knowledge of multi- plication, or the integral calculus rests upon the obser- vation and remembrance of the results of the differential calculus, so induction requires a piior knowledge of deduction. An inverse process is the undoing of the direct process. A person who enters a maze must either trust to chance to lead him out again, or he must carefully notice the road by which he entered. The facts furnished to us by experience are a maze of particular results; we might by chance observe in them the fulfilment of a law, but this is scarcely possible, unless we thoroughly learn the effects which would attach to any particular law. Accordingly, the importance of deductive reasoning is doubly supreme. Even when we gain the results of in- duction they would be of no use unless we could deduc- tively apply them. But before we can gain them at all we must understand deduction, since it is the inversion of deduction which constitutes induction. Our first task in this work, then, must be to trace out fully the nature of identity in all its forms of occurrence. Having given any series of propositions we must be prepared to develop deductively the whole meaning embodied in them, and the whole of the consequences which flow from them. il INTRODUCTION. IS >f Symbolic Expression of Logical Inference. In developing the results of the Principle of Inference we require to use an appropriate language of signs. It would indeed be quite possible to explain the processes of reasoning by the use of words found in the dictionary. Special examples of reasoning, too, may seem to be more readily apprehended than general symbolic forms. But it has been shown in the mathematical sciences that the attainment of truth depends greatly upon the invention of a clear, brief, and appropriate system of symbols. Not only is such a language convenient, but it is almost essential to the expression of those general truths which are the very soul of science. To apprehend the truth of special cases of inference does not constitute logic ; we must apprehend them as cases of more general ''truths. The object of all science is the separation of w^hat is common and general from what is accidental and different. In a system of logic, if anywhere, we should esteem this generality, and strive to exhibit clearly what is similar in very diverse cases. Hence the great value of general symbols by which we can represent the form of a reasoning process, disentangled from any consideration of the special subject to which it is applied. The signs required in logic are of a very simple kind As sameness or difference must exist between two things or notions, we need signs to indicate the things or notions compared, and other signs to denote the relations between them. We need, then, (i) symbols for terms, (2) a symbol for sameness, (3) a symbol for difference, and (4) one or two symbols to take the place of conjunctions. Urdmary nouns substantive, such as Iron, Metal, Elec- ^ incuy, Undulation, might serve as terms, but, for the ^ons explained above, it is better to adopt blank letters, devoid of special signification, such as A, B. C, Ac! Mch letter must be understood to represent a noun, and. 80 lar as the conditions of the argument allow, any noun, iZlfl^' '"^ ^^f"^'. \y> ^> V. ?, &c. are used for any El ' ?.^?^t«^\ed or unknown, except when the specml conditions of the problem are taken into account, 80^1 our letters stand for undetermined or unknown 14 THE PRINCIPLES OF SCIENCE. [oh AT '1 INTRODUCTION. Id :il 1 ! /.( f These letter-terms will be used indifferently for nouns substantive and adjective. Between these two kinds of nouns there may perhaps be differences in a metaphysical or grammatical point of view. But grammatical usa^re sanctions the conversion of adjectives into substantives and vice versd; we may avail ourselves of this latitude without in any way prejudging the metaphysical difficulties which may be involved. Here, as throughout tliis work I sliall devote my attention to truths which I can exhibit in a clear and formal manner, believing that in the present condition of logical science, this course will lead to areater advantage than discussion upon the metaphysical questions which may underlie any part of the subject. ^ Every noun or term denotes an object, and usually implies the possession by that object of certain qualities or circumstances common to all the objects denoted. There are certain terms, however, which imply the absence of qualities or circumstances attaching to other objects. It will be convement to employ a special mode of indicating these negative terms, as they are caUed. If the general name A denotes an object or class of objects possessing certain defined quaUties, then the term Not A will denote any object which does not possess the whole of those qualities ; m short. Not A is the sign for anything which dittere from A m regard to any one or more of the assigned qualities. If A denote " transparent object," Not A will denote "not transparent object." Brevity and facility of expression are of no slight importance in a system of notation, and it will therefore be desirable to substitute for the negative term Not A a briefer symbol. De Morean represented negative terms by small Roman letters or sometimes by small italic letters ;i as the latter seeiii to be highly convenient, I shall use a, J, c, . . . p, y. &c., aa the negative terms corresponding to A, B, C, . . . P, Q, &c. Thus if A means " fluid," a wiU mean " not fluid." Expression of IderUity and Difference, To denote the relation of sameness or identity I unhesi- tatingly adopt the sign =, so long used by mathematicians to denote equality. This symbol was originally appropriated * Formal Logic, p. 38. (I by Robert Recorde in his Whetstone 0/ Wit, to avoid the tedious repetition of the words "is equal to;" and he chose a pair of parallel lines, because no two things can bo more equal.^ The meaning of the sign has however been graduaUy extended beyond that of equaHty of quantities ; mathematicians have themselves used it to indicate equivalence of operations. The force of analogy has been so great that writers in most other branches of science have employed the same sign. The philologist uses it to indicate the equivalence of meaning of words : chemists adopt It to signify identity in kind and equality in weight of the elements which form two different compounds Not a few logicians, for instance Lambert, Drobitsch George Bentham,^ Boole,' have employed it as the copula of propositions. De Morgan declined to use it for this purpose, but still further extended its meaning so as to include the equivalence of a proposition with the premises from which it can be inferred ; * and Herbert Spencer has applied it m a like manner.*^ Many persons may think that the choice of a symbol is a matter of slight importance or of mere convenience : but 1 hold that the common use of this sign = in so many different meanings is reaUy founded upon a generalisation ot the widest character and of the greatest importance- one indeed which it is a principal pui-pose of this work to explain. The employment of the same sign in different cases would be unphilosophical unless there were some real analogy between its diverse meanings. If such analogy exists, It is not only aUowable, but highly desirable aSd even imperative, to use the symbol of equivalence with a generahty of meaning corresponding to the generality of the principles involved. Accordingly De Morgan's refusal to use the symbol in logical propositions indicated his opinion that there was a want of analogy between logical propositions and mathematical equations. I use the sign because I hold the contrary opinion. J flam's Literature of Europe, First Ed., vol. ii. p. 444. OuUtne of a New SysUm of Logic, London, 1827, ppTi, &. ; An Investtgation of tJu Laws of Thought,\p. 27, &i.^^' aJsZt f/T' PPV^^.'°^- .1° ^18 later wo?k, The Syllabus of a a^^jsUm of Logic, he discontinued the use of the sign ^ Principles of Ptychology, Second Ed., vol. il pp. 5^*55 Hi ;i II m '' ^h II le THE PRINCIPLES OF SCIENCE. [chip. I conceive that the sign =■ as commonly employed, always denotes some form or degree of sameness, and the particular form is usually indicated by the nature of the terms joined by it. Thus " 6,720 pounds = 3 tons " is evidently an equation of quantities. The formula — X — = + ex- presses the equivalence of operations. " Exogens = Dico- tyledons " is a logical identity expressing a profound truth concerning the character and origin of a most important group of plants. We have great need in logic of a distinct sign for the copula, because the little verb is (or are), hitherto used both in logic and ordinary discourse, is thoroughly am- biguous. It sometimes denotes identity, as in " St. Paul's is the chef-d'osuvre of Sir Christopher Wren ; " but it more commonly indicates inclusion of class within class, or partial identity, as in " Bishops are members of the House of Lords." This latter relation involves identity, but requires careful discrimination from simple identity, as will be shown further on. When with this sign of equality we join two nouns or logical terms, as in Hydrogen = The least dense element, we signify that the object or group of objects denoted by one term is identical with that denoted by the other, in everything except the names. The general formula A = B must be taken to mean that A and B are symbols for the same object or group of objects. This identity may some- times arise from the mere imposition of names, but it may also arise from the deepest laws of the constitution of nature ; as when we say Gravitating matter = Matter possessing inertia, Exogenous plants = Dicotyledonous plants, Plagihedral quartz crystals = Quartz crystals causing the plane of polarisation of light to rotate. We shall need carefully to distinguish between relations of terms which can be modified at our own will and those which are fixed as expressing the laws of nature ; but at present we are considering only the mode of expression which may be the same in either case. Sometimes, but much less frequently, we require a symbol to indicate difference or the absence of complete I.J INTRODUCTION. 17 sameness. For this purpose we may generalise in like manner the symbol -', which was introduced by Wallis to signify difference between quantities. The general formula B - C denotes that B and C are the names of two objects or groups whicli are not identical with each other. Thus we may say Acrogens ^ Flowering plants. Snowdon ^ The highest mountain in Great Britain. I shall also occasionally use the sign cos to signify in the most general manner the existence of any relation between the two terms connected by it. Thus c//i might mean not only the relations of equality or inequality, sameness or difference, but any special relation of time, place, size, causation, &c. in which one thing may stand to another. By A C05 B I mean, then, any two objects of thougl^ related to each other in any conceivable manner. General Formula of Logical Inference, The one supreme rule of inference consists, as I have said, in the direction to affirm of anything whatever is known of its like, equal or equivalent. The SiibstUution of Similars is a phrase which seems aptly to express the capacity of nmtual replacement existing in any two objects wJiich are like or equivalent to a sufficient degree. It is matter for furtlier investigation to ascertain when and for what purposes a degree of similarity less than complete identity is sufficient to warrant substitution. For the present we think only of the exact sameness expressed in the form A-B. Now if we take the letter to denote any third con- ceivable object, and use the sign c^ in its stated meaninc oi tndefimte relation, then the general formula of all inlerence may be thus exhibited :— ^rom A = B :<>: we may infer A g6» C or, m words— /w whatever relation a thing stands to a second thin^, in the same relation it stands to the like or eqmvalent of that second thing. The identity between A 18 THE PRINCIPLES OF SCIENCE. [CHAf. 1.1 INTRODUCTION. in and B allows us indififerently to place A where B was, or B where A was ; and there is no limit to the variety of special meanings which we can bestow upon the signs used in this fonnula consistently with its truth. Thus if we first specify only the meaning of the sign coj, we may say that if C is tlie weight of B, then G is also the weight of A. Similarly If C is the father of B, C is the father of A ; If C is a fragment of B, C is a fragment of A ; If C is a quality of B, C is a quality of A ; If C is a species of B, C is a species of A ; If C is the equal of B, C is the equal of A ; and so on ad infinitum. We may also endow with special meanings tlio letter- terms A, B, and C, and the process of inference will never be false. Thus let the sign ooo mean " is height of," and let A = Snowdon, B = Highest mountain in England or Wales, C = 3.590 feet; then it obviously follows since " 3,590 feet is the Height of Snowdon," and " Snowdon = the highest mountain iu England or Wales," that, " 3,590 feet is the height of the highest mountain in England or Wales." One result of this general process of inference is that we may in any aggregate or complex whole replace any part by its equivalent without altering the whole. To alter is to make a difference ; but if iu replacing a part I make no difference, there is no alteration of the whole. Many inferences which have been very imperfectly included in logical formulas at once follow. I remember the late Prof. De Moi-gan remarking that all Aristotle's logic could not prove that " Because a horse is an animal, the head of a horse is the head of an animal." I conceive that thia amounts merely to replacing in the complete notion head of a horse, the term " horse," by its equivalent some animal or an animal. Similarly, since The Lord Chancellor = The Speaker of the House of Lords, it follows that The death of the Lord Chancellor « The death of the Speaker of the House of Lords ; and any event, circumstance or thing, which stands iu a certain relation to the one will stand in like lelation to the other. Milton reasons in this way when he says, in his Areopagitica, " Who kills a man, kills a reasonable creature, God's image." If we may suppose him to mean G^d's image = man = some reasonable creature, it follows that " The killer of a man is the killer of some reasonable creature," and also " The killer of God's image.'- This replacement of equivalents may be repeated over and over again to any extent Thus if person is identical in meaning with individual, it follows that Meeting of persons = meeting of individuals ; and if assemblage = meeting, we may make a new 'replace- ment and show that Meeting of persons = assemblage of individuals. We may in fact found upon this principle of substitution a most genei-al axiom in the following terms ^ ; Same parts samely related mdce same wholes. If, for instance, exactly similar bricks and other materials be used to build two houses, and they be simi- larly placed in each house, the two houses must be similar. There are millions of cells in a human body, but if each cell of one person were represented by an exactly similar cell similarly placed in another body, the two persons would be undistinguishable, and would be only numerically different. It is upon this principle, as we shall see, that all accurate processes of measurement depend. If for a weight in a scale of a balance we substitute another weight, and the equilibrium remains entirely unchanged then the weights must be exactly equal. The general test of equality is substitution. Objects are equally bright when on replacing one by the other the eye perceives no difference. Objects are equal in dimensions .when tested by the same gauge they fit in the same manner. Generally speaking, two objects are alike so far as when substituted one for another no alteration is produced, and vice versd when alike no alteration is produced by the substitution. » Purt Tjogk, or the Logic of Quality^ p. 14. c 2 /I THE PRINCIPLES OF SCIENCE. fcifAf. The Propagating Power of Similarity, The relation of similarity in all its degrees is reciprocal So far as things are alike, either roay be substituted for the other; and this may perhaps be considered the very meaning of the relation. But it is well worth notice that there is in similarity a peculiar power of extending itself among all the things which are similar. To render a number of things similar to each other we need only render them similar to one standard object Each coin struck from a pair of dies not only resembles the matrix or original pattern from wliich the dies were struck, but resembles every other coin manufactured from the same original pattern. Among a million such coins there are not less than 499>999> 5 00,000 pairs of coins resembling each other. Similars to the same are similars to all. it is one great advantage of printing that all copies of a iocument struck from the same type are necessarily identical each with each, and whatever is true of one copy will be true of every copy. Similarly, if fifty rows of pipes in an organ be tuned in perfect unison with one row, usually the Principal, they must be in unison with each other. Similarity can also reproduce or propagate itself ad infinitum : for if a number of tuning-forks be adjusted in perfect unison with one standard fork, all instruments tuned to any one fork will agree with any instrument tuned to any other fork. Standard measures of length, capacity, weight, or any other measurable quality, are propagated in the same manner. So far as copies of the original standard, or copies of copies, or copies again of those copies, are accurately executed, they must all agree each with every other. It is the capability of mutual substitution which gives such great value to the modern methods of mechanical construction, according to which all the parts of a machine are exact facsimiles of a fixed pattern. The rifles used in the British army are consti-ucted on the American inter- changeable system, so that any part of any rifle can be substituted for the same part of another. A bullet fitting one rifle will fit all others of the same bore. Sir J. i.1 INTltODUCTION, 21 Whitworth has extended the same system to the screws and screw-bolts used m connecting together the parts of machmes, by estabbshing a series of standard screws. Anticipations of tJie Fnrmple of Substitution. In such a subject as logic it is hardly possible to put forth any opinions which have not been in some decree previously entertained. The germ at lea^t of every doctrine will be found m earHer wi-itei^, and novelty mus^ arise chiefly in the mode of harmonising and develop deas When I first employed the process and name 0I substiution m logic,^ I was led to do so from analogy with the familiar mathematical process of substituting for a symbol it« value as given in an equation. In writhig my hrst logical essay I had a most imperfect conception of the ^KhTwe^'n7^^Y '' ''^ P^^^^^«' -^ ' ^^^^^l as It they were of equal importance, a number of other laws which now seem to be but particular cases of the one general rule of substitution. My second essay, '^The Substitution of Similars " was written shortly after I had become aware o™' Jeat simplihcation which may be effected by a proper aS cation of the principle of substitution/ I w^not Then RT.r^^ ""'^ '^^ ^^^^ ^^^' ^h« C^erma^ Wian l7Ll"ir^V.'^^^^^^ ^f substitutiorand had used the word itself in forming a theory of* the syllogism. My imperfect acquaintance with the German pSnfT?L ^^?r^^'' but there is no doubt that othcr?oS.r '" "^^' '^ '"^^°- '^^^ ^^> ^^d probably other logicians, were m some degree famiUar with the principle * Even Aristotle's dictum manrreirarded modifv ^hif^- I ^^'^ P^'""^^ ^^^' w« have only to bn of fvf ^'^'''^. '''. ^^^^^da^ce with the quantifica^ tion of the predicate m order to arrive at the^ complete ' > 22 THE PRINCIPLES OF SCIENCE. [chap. 1.1 INTRODUCTION. M i, process of substitution.^ The Port-Royal logicians appear to have entertained nearly equivalent views, for they considered that all moods of the syllogism might bo reduced under one general principle.^ Of two premises they regard one as the containing proposition (propositio continens), and the other as the applicative proposition. The latter proposition must always be affirmative, and represents that by which a substitution is made; the former may or may not be negative, and is that in which a substitution is effected. They also show that this method will embrace certain cases of complex reason- ing which had no place in the Aristotelian syllogism. Their views probably constitute the greatest improvement in logical doctrine made up to that time since the days of Aristotle. But a true reform in logic must consist, not in explaining the syllogism in one way or another, but in doing away with all the narrow i-estrictions of the Aristotelian system, and in showing that there exists an infinite variety of logical arguments immediately deducible from the principle of substitution of which the ancient syllogism forms but a small and not even the most important part The Logic of Relatives. There is a difficult and important branch of logic which may be called the Logic of Relatives. If I argue, for instance, that because l)aniel Bernoulli was the son of John, and John the brother of James, therefore Daniel was the nephew of James, it is not possible to prove ttiis conclusion by any simple logical process. We re- quire at any rate to assume that the son of a brother is a nephew. A simple logical relation is that which exists between properties and circumstances of the same object or class. But objects and classes of objects may also be related according to all the properties of time and space. I believe it may be shown, indeed, that where an inference concerning such relations is drawn, a process of sub- stitution is really employed and an identity must exist ; * Svhstitution of Similars (1869), p. 9. • Port-Royal Logic, traDsl. by Spencer Baynes, pp. 212-219. Part III. chap. x. and xi 23 but I will not undei-take to prove the assertion in thia work. The relations of time and space are logical relations of a complicated character demanding much abstract and difficult investigation. The subject has been treated with such great ability by Peirce,^ De Morgan,* Ellis,^ and Harley, that I will not in the present work attempt any review of their writings, but merely refer the reader to the publications in which they are to be found. » Description of a Notation for the Logic of Relatives, resulting from an Am.phJicatton of the Conceptions of Boole's Calculus of Lofjic. By C. S. Peirce. Memoirs of the American Academy , vol. ix. Cam- bridge, U.S., 1870. 2 On the Syllogism No IV., and on tiie Logic of Relations. By Augustus De Morgan. Transactions of ths Cambridge FhUosophicaX Society, vol. x. part ii., i860. 3 Observations on Boole's Laws of Thought. By the Jate E. Leslie Ellis ; conimunicuted by the Rev. Robert Harley, F.R.S. Report of Vie British Association, 1870. Report of Sections, p. 12. Alio, On Books Laws of Thought. By the Rev. Robert Harley, F.R.S.. ibid. p. 14. J» f III \ , \ OBAF. IL] TERM& S6 ii CHAPTfiB II. TEBHS. »nf, T*r?^'''°°."P'T^ "»« resemblance or differ- ence of the things denoted by its tenns. As inference treats of the relation between two or more proposE s^ a proposition expresses a relation between two or more terms. In the portion of this work which t^te of Jeduction It will be convenient to follow the usu^^rder of exposition. We will consider in succession the va^Sus kinds of terms, propositions, and amuments and weTm meiice in this chapter with terms The simplest and most palpable meaning which can belong to a term consiste of s^me. single material obj^t such a^ Westminster Abbey. Stonehenge, the Sun. sS &c It IS probable that in early stagel of intellect 3 concrete and palpable things are^ th^ob/ecte S^Sughf h«^i«ry); ^"^ '^? 'ew'goise l>is master ammig a hundred other persons, and animals of much lower intel hgence kpow and discriminate their haunts iTdl such acts there is judgment concerning the likeness of XyS objecte, but there is little or no"" power of analysfnS object and regarding it as a group ^f qualities. ^ ^ The dignity of intellect begins with the power of separating points of agreement from those of dCncf Comparison of two objects may lead us to perceivrth^t they are at once like and unlike^ TwoTragmentTof ^ may differ entirely in outward form, yet tKay have^e same colour, hardness, and texture. Rowere^ych^^ m colour may differ in odour. The mind K to r^ each obj«5t as an aggregate of qualities, and acquires the power of dwelling at will upon one or otheTof those ^.rl'^T *^ .e^lus'on of the rest. Logical abstraction. m short, comes into play, and the mind becomes capable of reasoning, not merely about objects which are physically comp lete and concrete, but about things which may be thought ol sepai-ately in the mind though they exist not separately m nature We can think of "the hardness of L,^5f' °\-'' ?''''"' "f a flower, and thus produce abstract notions, denoted by abstract terms, which will form a subject for further consideration. nitl w°*® *""° *"*" S^'ieral notions and classes of- objects. We cannot fail to observe that the quality hard- "f* exists ,n many objects, for instance in many fr^m^nte clasTw 1*"^ n-g. the^e together, we^create The class Mrrf olyect, which will include, not only the actual sli^s i.!^"!!' V^^y ^* "^^^ ^^ «'^«'- ^ «ur s3wTrnn^.°^''''T, «P»rt to us aU the contents of oCts whXr!^ UBuaUy set any limits to the number of we wS„ iT^ ^^.u'""^ *"y ^'"'^ <=1^- At this point whi, jfl to perceive the power and generality of thought or i Tnfii^f ,"' "* " ''"Sle act to treat of'^indefinitely m' ! u \"'^°''*.^y numerous objects. We can safely assert eSlt'^"' « t'"e «f any 'one object coming Id^r a ctass IS true of any of the other objects so far as thev Tcl^''' W^'^'l"''^f '""P^^d L therbSonSnS ^ve^Xren]l^?''K r' ^Y? * ^^^S in a class uidess cla^^n^^^ 1^ }^^^y^ "^ ^' ^^ that is believed of the CO .Tideratirj^ ', ^h' ',' ""T^' " ""^'^"^ °f i«°PO^t-nt wiisiueration to decide how far and in what manner we can safely undertake thus to assign the place of StsTn WyTirr" °' '"^'^^ -^-^ constUuCh: Two/old Meaning of Oeneral Names. are^cST?J!f the "^nin^ of a name is that which we e™p«i "^^ °^ *h^° "'« "a^e is used. Now every thLkof «.^'''"8"'8toaclass; ^* ""^^ also cause us to '"".k of the common qualities possessed by those objects 26 THE PRINCIPLES OF SCIENCE. [criAP. II.J TERMS. 2t I* fi I A name is said to denote the object of tlioiiglit to which it may be applied ; it implies at the same time the possession of certain qualities or circumstances. The objects denoted form the extent of meaning of the term ; the qualities implied form the intent of meaning. Crystal is the name of any substance of which the molecules are arranged in a regular geometrical manner. The substances or objects in question form the extent of meaning ; the circumstance of having the molecules so arranged forms the intent of meaning. When we compare general terms together, it may often be found that the meaning of one is included in the mean- ing of another. Thus all crystals ai*e included among material substances^ and all opaque cin/stals are included among crystals; here the inclusion is in extension. We may also have inclusion of meaning in regard to intension. For, as all crystals are material substances, the qualities implied by the term material substance must be among those implied by crystal. Again, it is obvious that while in extension of meaning opaque crystals are but a pai*t of crystals, in intension of meaning crystal is but part of opaque crystal. We increase the intent of meaning of a term by joining to it adjectives, or phrases equivalent to adjectives, and the removal of such adjectives of course decreases the intensive meaning. Now, concerning such changes of meaning, the following all-important law holds universally true : — When the intent of meaning of a teim is increased the extent is decreased ; and vice versa, when the extent is increased the intent is decreased. In short, as one is increased the other is decreased. This law refers only to logical changes. The number of steam-engines in the world may be undergoing a rapid increase without the intensive meaning of the name being altered. The law will only be verified, again, when there is a real change in the intensive meaning, and an adjective may often be joined to a noun without making a change. Elementary metal is identical with metal; mortal man with man; it being a property of all metals to be elements, and of all men to be mortals. There is no limit to the amount of meaning which a term may have. A term may denote one object, or many, or an infinite number ' it mav im^ly a single quality, if such there be, or a group of any number of qualities, and vet the law connecting the extension and intension will in- faUibly apply. Taking the general name planet, we increase its intension and decrease its extension bv prefixing the adjective exteHcrr ; and if we further add nearest to tlu earth, there remains but one planet. Mars to which the name can then be applied. Singular terms which denote a single individual only, come under the same law of meaning as general names. They may be regarded as general names of which the meaning in exten- sion is reduced to a minimum. Logicians have erroneously asserted, as it seems to me, that singular t^rms are devoid of meaning m intension, the fact being that they exceed all other terms m that kind of meaning, as I have else- where tned to show.i Abstract Terms. Comparison of objects, and analysis of the complex resemblances and differences which they present, lead us to the conception of ahdract qimlUies. We learn to think of one object as not only different from another, but as dittering m some particular point, such as colour, or weight, or size. We may then convert points of agreement or difference into separate objects of thought which we call qualities and denote by abstract terms. Thus the terra .o rediuss means something in which a number of objects agree as to colour, and in virtue of which they are caUed red. Kedness forms, in fact, the intensive meanins of the term red. * Abstract terms are strongly distinguished from general terms by possessing only one kind of meaning; for ^ they denote qualities there is nothing which they cannot in Sri'"".? ^- ?^ l^J'"*^^" " ^ " i« ^^'^ ^^^^ of red Objects, but It implies the possession by them of the quality Sel afsoT S^^?!fr^^/^T T ^^' ??' ^'"^I P^re Logic, p. 6. Sheaden^^ llf / ^f r"^ ^^ f '^'i' ^^^'^ ^' ^^P' "' sections and giieddens Elements of Logic, London, 1864, pp 14. &c Profpssr.r Robertson objects {Mind,\o\, i, p. 2ii) thatPco^se^uLf ^nd pro^^ names ; if so it « because I hold that the same i-emf rks app^y to proi^r names, which do not seem to me to differ lo^caUv fiSm •uif^iar names. i m 28 THE PRINCIPLES OP SCIENCE. [chap. redness ; but this latter term has one single meaning— the quality alone. Thus it arises that abstract terms are in- capadle of plurality. Eed objects are numerically distinct each from each, and there are multitudes of such objects • but redness is a single quality which runs through all those objects, and is the same in one as it is in another It IS true that we may speak of rednesses, meaning different kmds or tints of redness, just as we may speak of colours. meaning different kinds of colours. But in distinguLshinJ kinds, degrees, or other differences, we render the terms so tar concrete. In that they are merely red there is but a single nature m red objects, and so far as things are merely coloured, colour is a single indivisible quality. Redness, so far as it is redness merely, is one and the same every- where, and possesses absolute oneness. In virtue of this unity we acqiure the power of treating all instances of such quabty as we may treat any one. We possess in short, genei-al knowledga Substantial Terms, Logicians appear to have taken little notice of a class of terms which partake in certain respects of the character of abstract terms and yet are undoubtedly the names of con- crete existing things. These terms are the names of substances, such as gold, carbonate of lime, nitrogen &c We cannot speak of two golds, twenty carbonates of lime' or a hundred nitrogens. There is no such distinction between the parts of a uniform substance as will allow of a discrimination of numerous individuals. The qualities of colour, lustre, malleability, density, &c., by which we recogmse gold, extend through its substance irrespective of particular size or shape. So far as a substance is gold it is one and the same everywhere ; so that terms of this kind, which I propose to caU substantial terms, possess the peculiar unity of abstract terms. Yet they are not abstract; for gold is of course a tangible visible body ^tirely concrete, and existing independently of othe^ It is only when, by actual mechanical division, we break up the uniform whole which forms the meaning of a substantial term, that we introduce number. Piece of gold ti.j TERMS. Si» is a term capable of plurality ; for there may be a great many pieces discriminated either by their various shapes and sizes, or, in the absence of such marks, by simul- taneously occupying different parts of space. In substance they are one ; as regards the properties of space they are many.i We need not further pursue this question, which involves the distinction between unity and plurality, until we consider the principles of number in a subsequent chapter. Collective Terms. We must clearly distinguish between the collective and the general meanings of terms. The same name may be used to denote the whole body of existing objects of a certain kind, or any one of those objects taken separately. " Man " may mean the aggregate of existing men, which we sometimes describe as mankind; it is also the general name applying to any man. The vegetable kingdom is the name of the whole aggregate of plants, but " plant " itself is a general name applying to any one or other plant. Every material object may be conceived as divisible into parts, and is therefore collective as regards those parts. Tlie animal body is made up of cells and fibres, a crystal of molecules; wherever physical division, or as it has been called partition, is possible, there we deal in reality with a collective whola Thus the greater number of general terms are at the same time collective as regards each individual whole which they denote. It need hardly be pointed out that we must not infer of a collective whole what we know only of the parts, nor of the parts what we know only of the whole. The relation of whole and part is not one of identity, and does not allow of substitution. There may nevertheless be qualities which are true alike of the whole and of its parts. A number of organ-pipes tuned in unison produce an aggre- gate of sound which is of exactly the same pitch as each * Professor Robertson has criticised my introduction of "Substantial l.erms {Mind, vol. i. p. 210), and objects, perhaps correctly, that the aistinction if valid is extra-logical. I am inclined to think, however, uiat the doctrine of terms ia, strictly speaking, for the most part C7 30 W- !i ' i I i THE PRINCIPLES OF SCIENCE. [CUAP. separate. sound. In the case of substantial terms, certain qualities may be present equally in each minutest part as in the whole. The chemical nature of the largest mass of pure carbonate of lime is the same as the nature of the smallest particle. In the case of abstract terms, again, we cannot di-aw a distinction between whole and part ; what ia true of redness in any case is always true of redness, so far as it is merely red. ^ Synthesis of TenM, We continually combine simple terms together so as to form new terms of more complex meaning. Thus, to increase the intension of meaning of a term we write it with an adjective or a phrase of adjectival nature. By joining "brittle" to "metal," we obtain a combined term, "brittle metal," which denotes a certain portion of the metals, namely, such as are selected on account of pos- sessing the quality of brittleness. As we have already seen, " brittle metal " possesses less extension and greater intension than metal. Nouns, prepositional phrases, parti- cipial phrases and subordinate propositions may also be added to terms so as to increase their intension and decrease their extension. In our symbolic language we need some mode of indi- cating this junction of terms, and the most convenient device will be the juxtaposition of the letter-terms. Thus if A mean brittle, and B mean metal, then AB will mean brittle metal Nor need there be any limit to the number of letters thus joined together, or the complexity of the notions which they may represent. Thus if we take the letters P = metal, Q = white, R = monovalent, S = of specific gravity iO'5, T = melting above 1000° C, V = good conductor of heat and electricity, then we can form a combined term PQRSTV, which will denote "a whit« monovalent metal, of specific gravity 10 5, melting above 1000° C, and a good conductor of heat and electricity." II.] TEKMS. SI There are many grammatical usages concerning tho junction of words and phrases to which we need pay no attention m logic. We can never say in ordinary languafje "of wood table,' meaning "table of wood;" but we may consider "of wood" as logically an exact equivalent of " wooden ; so that if X ~ of wood, Y = table, there is no reason why, in our symbols, XY should not be just as correct an expression for " table of wood " as YX In this case indeed we might substitute for "of wood " the correspondmg adjective " wooden," but we should often fail to find any adjective answering exactly to a phrase. There is no single word by which we could express the notion 'of specific gravity 105 : " but logically we may consider these words as forming an adjective; and denoting this by b and metal by P, we may say that SP means " metal of specific gmvity 105." It is one of many advantages in these blank letter-symbols that they enable us completely to neglect all grammatical peculiarities and to fix our attention solely on the purely logical relations involved Investigation will probably show that the rules of grammar are mainly founded upon traditional usage and have little ogical signification. This indeed is sufficiently proved by tlie wide grammatical differences which exist between languages, though the logical foundation must be the same. Syniholic Expression of the Law of Contradiction. T '^'^^VXS^'^®^^® ^^ ^™^ ^^ subject to the all-important law of Thought, described in a previous section (p. c) and called the Law of Contradiction. It is self-evident that no quality can be both present and absent at the same time and place. This fundamental condition of all thought and ot aU existence is expressed symbolically by a rule that a terni and its negative shall never be allowed to come into combination. Such combined terms as Aa, Bb, Cc, &c are seii-contradictory and devoid of all inteUigible meaning It they could represent anything, it would be what cannot l^nU r '^''''^ ^""^'^ ^^ imagined in the mind. They can therefore only ent^r into our consideration to suffer :^ r i\ 32 THE PRINCIPLES OF SCIENCK [chap. immediate exclusion. The criterion of false reasoning, as we shall find, is that it involves self-contradiction, the affirm- ing and denying of the same statement. We might repre- sent tne object of all reasoning as the separation of the consistent and possible from the inconsistent and impossi- ble ; and we cannot make any statement except a truism without implying that certain combinations of terms are contradictory and excluded from thought. To assert that " all A's are B's " is equivalent to the assertion that " A's which are not B's cannot exist." It will be convenient to have the means of indicating the exclusion of the self-contradictory, and we may use the familiar sign for nothing, the cipher o. Thus the second law of thought may be symbolised in the forms Aa = o ABh = o ABCa = o We may variously describe the meaning of o in logic as the non-existejit, the impossible, the self-incoumtent, the inconceivable. Close analogy exists between this meaning and its mathematical signification. Certain Special Conditions of Logical Symbols. In order that we may argue and infer truly we must treat our logical symbols according to the fundamental laws of Identity and Difference. But in thus using our symbols we shall frequently meet with combinations of which the meaning will not at first sight be apparent If in one case we learn that an object is " yellow and round," and in another case that it is " round and yellow," there arises the question whether these two descriptions are identical in meaning or not. Again, if we proved that an object was " round round," the meaning of such an expres- sion would be open to doubt. Accordingly we must take notice, before proceeding further, of certain special laws which govern the combination of logical terms. In the first place the combination of a logical term with itself is without effect, just as the repetition of a statement does not alter the meaning of the statement ; " a round round object" is simply "a round object." What is yellow yellow is merely yellow; metallic metals cannot differ from metals, nor circular circles from circles. In nu/ I II.] TERMS. n symbolic language we may similarly hold that A A is iden- tical with A, or A = AA = AAA = &c. The late Professor Boole is the only logician in modern times who has drawn attention to this remarkable property of logical terms ; ^ but in place of the name which he gave to the law, I have proposed to call it The Law of Simpli- city.* Its high importance will only become apparent when we attempt to determine the relations of logical and mathematical science. Two symbols of quantity, and only two, seem to obey this law ; we may say that i x i = i, and 0x0 = (taking o to mean absolute zero or i - i) ,' there is apparently no other number which combined with itself gives an unclianged result. I shall point out, how- ever, in the chapter upon Number, that in realitv all numerical symbols obey this logical principle. It is curious that this Law of Simplicity, though almost unnoticed in modern times, was known to Boethius, who makes . a singular remark in his treatise De Trinit'ate et Unitate Dei (p. 959). He says : *• If I should say sun, sun, sun, I should not have made three suns, but I should have named one sun so many times." » Ancient discussions about the doctrine of the Trinity drew more attention to subtle questions concerning the nature of unity and plui-ality than has ever since been given to them. It is a second law of logical symbols that order of com- bmation is a matter of indifference. " Rich and rare gems " are the same as " rare and rich gems," or even as " gems, rich and rare." Grammatical, rhetorical, or poetic usage may give considerable significance to order of expression. The limited power of our minds prevents our grasping many ideas at once, and thus the order of statement may produce some effect, but not in a simply logical manner. All life proceeds in the succession of time, and we are obliged to write, speak, or even think of things and their qualities one after the other ; but between the things and tneir qualities there need be no such relation of order in T^^^^^f^*^^ ^wa/ym 0/ Logic, Cambridge, 1847, p. 17. An Jnte^atton of the Latoa of Thought, London, iSsi p. 31. ^ ^ ^ rure Logic, p. 15. totie«'p^icale^' ^^' ^^ ^''^' ''''° ^ "^^"^ ^^'''""'' *''^ ^''^ b ^ THE PRINCIPLES OF SCIENCE. [caAP u.) TERMS. » 'J iH' 1 ■ I w time or space. The sweetness of sugar is neither before nor after its weight and solubility. Tlie hardness of a metal, its colour, weight, opacity, malleability, electric and chemical properties, are all coexistent and coextensive, per- vading the metal and every part of it in perfect community, none before nor after the others. In our words and symbols we cannot observe this natural condition ; we must name one quality first and another second, just as some one must be the first to sign a petition, or to walk foremost in a pro- cession. In nature there is no such precedence. I find that the opinion here stated, to the effect that relations of space and time do not apply to many of our ideas, is clearly adopted by Hume in his celebrated Trea- tise on Human Nature (vol. i. p^ 410). He says :* — " An object may be said to be no where, when its parts are not so situated with respect to each other, as to form any figure or quantity ; nor the whole with respect to other bodies so as to answer to our notions of contiguity or distance. Now this is evidently the case with all our perceptions and objects, except those of sight and feeling. A moral reflection cannot be placed on the right hand or on the left hand of a passion, nor can a smell or sound be either of a circular or a square figure. These objects and perceptions, so far from requiring any particular place, are absolutely incom- patible with it, and even the imagination cannot attribute it to them." A little reflection will show that knowledge in the highest perfection would consist in the simultamaus pos- session of a multitude of facts. To comprehend a science perfectly we should have every fact present with every other fact. We must write a book and we must read it successively word by word, but how infinitely higher would be our powers of thought if we could grasp the whole in one collective act of consciousness ! Compared with the brutes we do possess some slight approximation to such power, and it is conceivable that in the indefinite future mind may acquire an increase of capacity, and be less restricted to the piecemeal examination of a subject. Bat I wish here to make plain that there is no logical foundation for the successive character of thought and reasoning unavoidable under our present mental conditions. ' Book i., Part it., Section 8. We are logically weak and imperfect in reject of tlve fact thai we are obliged to think of one thing after another. We must describe metal as " hard and opaque," or " opaque and hard," but in the metal itself there is no such difference of order ; the properties are simultaneous and coextensive in existence. Setting aside all grammatical peculiarities which render a substantive less moveable than an adjective, and dis- regarding any meaning indicated by emphasis or marked order of words, we may state, as a general law of logic, that AB is identical with BA, or AB = BA. Similarly, ABC = ACB = BCA = &c. Boole first drew attention in recent years to this pro- perty of logical terms, and he called it the property of Commutativeness.^ He not only stated the law with the utmost clearness, but pointed out that it is a Law of Thought rather than a Law of Things. I shall have in various parts of this work to show how the necessary im- perfection of our symbols expressed in this law clings to our modes of expression, and introduces complication into the whole body of mathematical formulae, which are really founded on a logical basis. It is of course apparent that the power of commutation belongs only to terms related in the simple logical mode of synthesis. No one can confuse " a house of bricks" with " bricks of a house," " twelve square feet " with " twelve feet square," "the water of crystallization" with '* the crystalliza- tion of water." All relations which involve differences of time and space are inconvertible ; the higher must not be made to • change places with the lower, nor the first with the last. For the parties concerned there is all the difference in the world between A killing B and B kiUing A. The law of com- mutativeness simply asserts that difference of order does not attech to the connection between the properties and circumstances of a thing— to what I call simple logical relation. fj ^^if^/ '^^^*^^^y P- 2Q. It is pointed out in the preface to this Second Edition that Leibnitz was acquainted with the Laws oS Simplicity and of Commutativeneaa, r 1 OHAP. III.] PROPOSITIONS. sr ! >i ('; K CHAFfER IIL PROPOSITIONS. We now proceed to consider the variety of fonns of pro- positions in which the truths of science must be expressed. I shall endeavour to show that, however diverse these forms may be, they all admit the application of the one same principle of inference that what is true of a thing is true of the like or same. This principle holds true what- ever be the kind or manner of the likeness, provided proper regard be had to its nature. Propositions may assert an identity of time, space, manner, quantity, degree, or any other circumstance in which things may agree or ilifPer. We find an instance of a proposition concerning time in the following : — " The year in which Newton was born, was the year in which Galileo died." This proposition expresses an approximate identity of time between two events; hence whatever is true of the year in which Galileo died is true of that in which Newton was born, and vice versd, " Tower Hill is the place where Raleigh was executed " expresses an identity of place ; and what- ever is true of the one spot is true of the spot otherwise defined, but in reality the same. In ordinary language we have many propositions obscurely expressing identities of number, quantity, or degree. " So many men, so many minds," is a proposition concerning number, that is to say, an equation; whatever is true of the number of men is true of the number of -minds, and vice versd. " The density of Mars is (nearly) the same as that of the Earth," " The force of gravity is directly as the product of the masses, and inversely as the square of the distance," are propositions concerning magnitude or degree. Logicians have not paid adequate attention to the great variety of propositions which can be stated by the use of the little conjunction as, together with so. " As the home so the people," is a proposition expressing identity of manner; and a great number of similar propositions all indicating some kind of resemblance might be quoted. Whatever be the special kind of identity, all such expressions are subject to the great principle of inference ; but as we shall in later parts of this work treat more particularly of inference in cases of number and magnitude, we will here confine our attention to logical propositions which involve only notions of quality. Simple IdeiUities, The most important class of propositions consists ofc those which fall under the formula \ A = B, and may be called simple identities. I may instance, in the first place, those most elementary propositions which express the exact similarity of a quality encountered in two or more objects. I may compare the colour of the Pacific Ocean with that of the Atlantic, and declare them identical. I may assert that '* the smell of a rotten ^gg is like that of hydrogen sulphide ; " " the taste of silver hypo- sulphite is like that of cane sugar ; " " the sound of an earthquake resembles that of distant artillery." Such are propositions stating, accurately or otherwise, the identity of simple physical sensations. Judgments of this kind are necessarily pre-supposed in more complex, judgments. If I declare that " this coin is made of gold," I must base the judgment upon the exact likeness of the substance in several qualities to other pieces of substance which are undoubtedly gold. I must make judgments of the colour, the specific gravity, the hardness, and of other mechanicaJ and chemical properties ; each of these judgments is ex pressed in an elementary proposition, " the colour of this coin is the colour of gold," and so on. Even when we establish the ide'itity of a thing with itself under a dilferent name or aspect, it i* by distinct judgments ,{ ^\ ilif (; ' j\| ^ f ! ;i I S6 THE PRINCIPLES OF SCIENCE. [CHAF. concerning single circumstances. To prove that the Homeric x"'^^^^ is copper we must show the identity of each quality rticorded of ^oXko^ with a quality of copper. To establish Deal as the landing-place of Caesar, all material circumstances must be shown to agree. If the modern Wroxeter is the ancient Uriconium, there must be the like agreement of all features of the country not subject to alteratio3i by time. Such identities must be expressed in the form A = B. We may say Colour of Pacific Ocean = Colour of Atlantic Ocean. Smell of rotten egg = Smell of hydro^^en sulphide. In these and similar propositions we assert identity of single qualities or causes of sensation. In the same form we may also express identity of any gi-oup of qualities, as ill X<i^fco<i = Copper. Deal = Landing-place of Caesar. A multitude of propositions involving singular terms fall into the same form, as in The Pole star = The slowest-moving star. Jupiter = The greatest of the planets. The ringed planet = The planet having seven satel- lites. The Queen of England = The Empress of India. The number two = The even prime number. Honesty = The best policy. In mathematical and scientific theories we often meet with simple identities capable of expression in the same form. Thus in mechanical science " The process for finding the resultsmt of forces = the process for finding the re- sultant of simultaneous velocities." Theorems in geometry often give results in this form, as Equilateral triangles = Equiangular triangles. Circle = Finite plane curve of constant curvature. Circle = Curve of least perimeter. The more profound and important laws of nature are often expressible in the form of simple identities; in addition to some instances which have already been given, I may suggest, Crystals of cubical system = Crystals not possessing the power of double refiactiou. III.] PROPOSITIONS 38 All definitions are necessarily of this form, whether the objects defined be many, few, or singular. Thus we may say, Common salt = Sodium chloride. Chlorophyl = Green colouring matter of leaves. Square = Equal-sided rectangle. It is an extraordinary fact that propositions of this elementary form, all-important and very numerous as they are, had no recognised place in Aristotle's system of Logic. Accordingly their importance was overlooked until verv recent times> and logic was the most deformed of sciences. But it is impossible that Aristotle or any other person should avoid constantly using them ; not a term could be defined without their use. In one place at least Aristotle actually notices a proposition of the kind. He observes • " We sometimes say that that white thing is Socrates, or that the object approaching is (/allias."^ Here we certainly have simple identity of terms ; but he considered such propositions purely accidental, and came to the unfoitunate conclusion, that " Singulars cannot be predicated of other terms." Propositions may also express the identity of extensive groups of objects taken collectively or in one connected whole ; as when we say, The Queen, I^rds, and Commons = The Legislature of the United Kingdom. When Blackstone asserts that " The only true and natural foundation of society are the wants and fears of individuals," we must interpret him as meaning that the whole of the wants and fears of individuals in the aggregate form the foundation of society. But many propositions which might seem to be collective are but groups of singular propositions or identities. When we say " Potassium and sodium are the metallic bases of potash and soda," we obviously mean. Potassium = Metallic base of potash ; Sodium = Metallic base of soda. It is the work of grammatical analysis to separate the various propositions often combined into a single sentence Logic cannot be properly required to interpret the forms and devices of language, but only to treat the meaning when clearly exhibited. Prior Analytics^ L cap. xxvii. v I/' I'} f 40 THE PRINCIPLES OF SCIENCE. [CHAF. ill.] PROPOSITIONS. 41 )0' 1 1 if \ 4 f 1,1 .1 rj Partial Identities. A second highly important kind of proposition is that which I propose to call a partial identity. When we say that "All mammalia are vertebrata," we do not mean that mammalian animals are identical with vertebrate animals, but only that the mammalia form a part of the class verte- hrata. Such a proposition was regarded in the old logic as asserting the inclusion of one class in another, or of an object in a class. It was called a universal affirmative pro- position, because the attribute vertebrate was affirmed of the whole subject mammalia ; but the attribute was said to be undistrihUedf because not all vertebrata were of necessity involved in the proposition. Aristotle, overlooking the im- portance of simple identities, and indeed almost denying their existence, unfortunately founded his system upon the notion of inclusion in a class, instead of adopting the basis of identity. He regarded inference as resting upon the rule that what is true of the containing class is true of the contained, in place of the vastly more general rule that what is true of a class or thing is true of the like. Thus he not only reduced logic to a fragment of its proper self, but destroyed the deep analogies which bind together logical and mathematical reasoning. Hence a crowd of defects, difficulties and errors which will long disfigure the first and simplest of the sciences. It is surely evident that the relation of inclusion rests upon the relation of identity. Mammalian animals cannot be included among vertebrates unless they be identical with part of the vertebrates. Cabinet Ministers are included almost always in the class Members of Parliament, because they are identical with some who sit in Parliament. We may indicate this identity with a part of the larger class in various ways ; as for instance, Mammalia = part of the vertebrata. Diatomace8e = a class of plants. Cabinet Ministers = some members of Parliament. Iron = a metal. In ordinary language the verbs is and are express mere inclusion more often than not. Men are mortals, means I that men form a part of the class mortal ; but great con- fusion exists between this sense of the verb and that in which it expresses identity, as in " The sun is the centre of the planetary system." The introduction of the indefinite article a often expresses partiality ; when we say " Iron is a metal" we clearly mean that iron is one only of several metals. Certain recent logicians have proposed to avoid the indefiniteness in question by what is called the Quanti- fication of the Predicate, and they have generally used the -^little word some to show that only a part of the predicate is identical with the subject Some is an indeterminate adjedive ; it implies unknown qualities by which we might select the part in question if the qualities were known, but it gives no hint as to their nature. I might make use of such an indeterminate sign to express partial identities in this work. Thus, taking the special symbol V = Some, the general form of a partial identity would be A = VB, and in Boole's Logic expressions of the kind were much used. But I believe that indeterminate symbols only introduce complexity, and destroy the beauty and simple universality of the system which may be created without their use. A vague word like some is only used in ordinary language by ellipsis, and to avoid the trouble of attaining accuracy.' We can always employ more definite expressions if we hke; but when once the indefinite some is introduced we cannot replace it by the special description. We do not know whether »ome colour is red, yellow, blue, or what it is ; but on the other hand red colour is certainly some colour. Throughout this system of logic I shall dispense with such indefinite expressions ; and this can readily be done by substituting one of the other terms. To express the proposition " All A's are some B's " I shall nou use the form A = VB. but A = AB. This formula states that the class A is identical with the class AB ; and as the latter must be a part at least of the class B, it implies the inclusion of the class A in that of B. We might represent our former example thus. Mammalia =s Mammalian vertebrata. This proposition asserts identity between a part (or it may 42 THE PRINCIPLES OF SCIENCE. [CHAP. III.] PROPOSITIONS. 1 m ^m 1 : be the whole) of the vertebrata and the mammalia. If it is asked What part ? the proposition affords no answer, except that it is the part which is mammalian ; but the assertion " mammalia = some vertebrata " tells us no mora It is quite likely that some readers will think this mode of representing the universal affirmative proposition artificial and complicated. I will not undertake to con- vince them of the opposite at this point of my exposition. Justification for it will be found, not so much in the im- mediate treatment of this proposition, as in the general harmony which it will enable us to disclose between all parts of reasoning. I have no doubt that this is the critical difficulty in the relation of logical to other forms of reasoning. Grant this mode of denoting that " all A's are B*s," and I fear no further difficulties ; refuse it, and we find want of analogy and endless anomaly in every direction. It is on general grounds that I hope to show overwhelming reasons for seeking to reduce every kind of proposition to the form of an identity. I may add that not a few logicians have accepted this view of the universal affirmative proposition. Leibnitz, in his IHficultates Qucedam Logicce^ adopts it, saying, " Omne A est B ; id est {equivalent AB et A, seu A non B est non- ens." Boole employed the logical equation x = xy con- currently with x = vy; and Spalding ^ distinctly says that the proposition " all metals are minerals " might be de- scribed as an assertion of partial identity between the two classes. Hence the name which I have adopted for the proposition. ^ , Limited Identities, An important class ot propositions have the form AB = AC, expressing the identity of the class AB with the class AC. In other words, " Within the sphere of the class A, all the B's are all the C's ; " or again, " The B's and C's, which are A*s, are identical." But it will be observed that nothiug is asserted concerning things which are outside of the class A ; and thus the identity is of limited extent. It is the proposition B = C limited to the sphere of things called A. > Encyclaptzdia Britannicaf Eighth Ed. art. Logic, sect. 37, note. 8vo reprint, p. 79. Thus we may say, with some appi-oximation to truth, that " Large plants are plants devoid of locomotive power." A barrister may make numbers of most general state- ments concerning the relations of persons and things in the course of an argument, but it is of course to be understood that he speaks only of persons and things under the English Iaw. Even mathematicians make statements which are not true with absolute generality. They say that imaginary roots enter into equations by pairs ; but this is only true under the tacit condition that the equations in question shall not have imaginary coefficients.^ The uni- verse, in short, within which they habitually discourse is that of equations with real coefficients. These implied limitations form part of that great mass of tacit knowledge which accompanies all special arguments. To Do Morgan is due the remark, that we do usually think and argue in a limited universe or sphere of notions, even when it is not expressly stated.* It is worthy of inquiry whether all identities are not really limited to an implied sphere of meaning. When we make such a plain statement as " Gold is malleable " we obviously speak of gold only in its solid state ; Avhen we say that " Mercury is a liquid metal " we must be under- stood to exclude the frozen condition to which it may be reduced in the Arctic regions. Even when we take such a fundamental law of nature as "All substances gravitate," we must mean by substance, material substance, not in- cludmg that basis of heat, light, and electrical undulations which occupies space and possesses many wonderful me- chanical properties, but not gravity. The proposition then is really of the form Material substance = Material gravitating substance. Negative Propositions. In every act of intellect we are engaged with a certain laentity or difference between things or sensations compared vB',- ^^^*^herto I have treated only of identities ; and yec It might seem that the relation of difference must be t^ph^l Tr^"^*-^ the Root of any Function. Cambridge Philo- ^nical Transactions, 1867, vol xi. p. 25. ^ ^Uabui of a propoted SytUm of Logic, §§ 122, 123. s 44 THE PRINCIPLES OF SCIENCE. (chap. infinitely more common than that of likeness. One thing may resemble a great many other things, but then it diflfers from all remaining things in the world. Diversity may almost be said to constitute life, being to thought what motion is to a river. The perception of an object involves its discrimination from all other objects. But we may nevertheless be said to detect resemblance as often as we detect difference. We cannot, in fact, assert the existence of a difference, without at the same time implying the existence of an agreement. If I compare mercury, for instance, with other metals, and decide that it is not solid, here is a difference between mercury and solid things, expressed in a negative propo- sition ; but there must be implied, at the same time, an agreement between mercury and the other substances which are not solid. As it is impossible to separate the vowels of the alphabet from the consonants without at the same time separating the consonants from* the vowels, so I cannot select as the object of thought solid things, without thereby throwing together into another class all things which are not solid. The very fact of not possessing a quality, constitutes a new quality which may be the ground of judgment and classification. In this point of view, agreement and difference are ever the two sides of the same act of intellect, and it becomes equally possible to express the same judgment in the one or other aspect Between atfii-mation and negation there is accordingly a perfect equilibrium. Every affirmative proposition implies a negative one, and vice versd. It is even a matter of in- difference, in a logical point of view, whether a positive or negative term be used to denote a given quality and the class of things possessing it. If the ordinary state of a man's body be called good health, then in other circumstances he is said not to he in good health ; but we might equally describe him in the latter state as sickly, and in his normal condition he would be not sickly. Animal and vegetable substances are now called organic, so that the other sub- stances, forming an immensely greater part of the globe, are described negatively as inorganic. But we might, with at least equal logical correctness, have described the prepon- derating class of substances as mineral, and then vegetable and animal substances would have been non-mineral. III.J PE0P0SITI0N8. 46 It is plam that any positive term and its corresponding negative divide between them the whole universe of thought : whatever does not fall into one must fall into the other, by the third fundamental Law of Thought, the Law of Duality. It follows at once that there are two modes of representing a difference. Supposing that the things represented by A and B are found to differ, we may indicate (see p. 17) the result of the judgment by the notation A *- B. We may now represent the same judgment by the assertion that A agrees with those things which differ from B or that A agrees with the not-B's. Using our notation 'for negative terms (see p. 14), we obtain A^Ab as the expression of the ordinary negative proposition. Thus if we take A to mean quicksilver, and B solid, then we have the following proposition : Quicksilver = Quicksilver not-solid There may also be several other classes of negative pro- positions, of which no notice wa^ taken in the old locric We may have cases where all A's are not-B's, and at the same time all not-B's are A's; there may, in short, be a simple identity between A and not-B, which ma^ be expressed m the form ^ A , -^ = ^• An example of this form would be Wn , ^e^f^ctora of electricity = non-electiics. duchon ^L ? ^'T^""^^^ l"^^^ ^^ ^^^^ ^ results of de- duction, with simple, partial, or limited identities between negative terms, as in the forms oeDween Tf lA^^^* a = <a), aC = 5C, etc. It would be possible to represent affirmative pronositions in the negative form. Thus '^ron is solid,'' m^Tbe^^^ orTalln'' 7"^^ ^ r ' ^^^«^^^^>" ^^ " IroA is nof fluid » tTe&i^b?/^: 'i^ ^^ "-n," and "not-soHd," alUroS^^^^^ very strong reasons why we should employ prSbvthp«\\^f'.^^'?^^^^ ^^™- ^U inference tKpreLd in ?^^^^^ equivalents, and a proposi- aU \fTFr^ ^^^ ^^™ 0^ ^ identity is ready to yield luiiy shown, we can infer in a negative proposition, 46 THE PRINCIPLBS OP SCIENOE. [OHAP. Ui.J PllOPOSITIONS. n but not by it. Difference is incapable of becoming tlie ground of inference ; it is only the implied agreement with other differing objects which admits of deductive reason- ing; and it will always be found advantageous to employ propositions in the form which exhibits clearly the implied agreements. Conversion of Propositions. The old books of logic contain many rules concerning the conversion of propositions, that is, the transposition ol" the subject and predicate in such a way as to obtain a new proposition which will be true when the original proposi- tion is true. The reduction of every proposition to the form of an identity renders all such rules and processes needless. Identity is essentially reciprocal. If the colour of the Atlantic Ocean is the same as that of the Pacific Ocean, that of the Pacific must be the same as that of the Atlantic. Sodium chloride being identical with common salt, common salt must be identical with sodium chlorida If the number of windows in Salisbury Cathedral equals the number of days in the year, the number of days in the year must equal the number of the windows. Lord Chesterfield was not wrong when he said, *'I will give anybody their choice of these two truths, which amount to the same thing ; He who loves himself best is the honestest man; or, The honestest man loves himself best" Scotus Erigena exactly expresses this reciprocal character of identity in saying, "There are not two studies, one of philosophy and the other of religion ; true philosophy is true religion, and true religion is true philosophy." A mathematician would not think it worth while to mention that if a: = y then also y = a;. He would not con- sider these to be two equations at all, but one equation accidentally written in two different manners. In written symbols one of two names must come first, and the other second, and a like succession must perhaps be observed in our thoughts: but in the relation of identity. there is no need for succession in order (see p. 33) -, each is simul- taneously equal and identical to the other. These remarks will hold true both of logical and mathematical identity ; 6? that I shall consider the two forms 47 A = B and B = A ^ to express exactly the same identity differently written All need for rules of conversion disappears, and there will be no single proposition in the system which mav not be written with either end foremost. Thus A = AB is the same as AB = A, aC = bG is the same as &C = aC and so forth. ' The same remarks are partially true of differences and inequalities, which are also reciprocal to the extent that one thing cannot differ from a second without the second diffenng from the first. Mars differs in colour from Venus, and Venus must differ from Mars. The Earth differs from Jupiter m density ; therefore Jupiter must differ from the ^rth Speaking generally, if A «- B we shaU also have B - A and these two forms may be considered ex- pressions of the same difference. But the relation of differing thmgs is not wholly reciprocal. The density of Jupiter does not differ from that of the Earth in the same way that that of the Earth differs from that of Jupiter The change of sensation which we experience in parsing from Venus to Mars is not the same as what we experience m passing bwjk to Venus, but just the opposite in nature. The colour of the sky is lighter than that of the ocean ; therefore that of the ocean cannot be lighter than that of the sky, but darker. In these and all similar cases we gain a notion of direction or character of change, and resulte of immense importance may be shown to rest on this, notion, ^or the present we shall be concerned with the mere fact ot Identity existing or not existing. Two/old Interpretation of Propositions. ^IV"^^' ^ "^^ ^^""^ ^^^"^ (P- 25), may have a meaning either in extension or intension ; and according as one cS the other meamng is attributed to the terms of a proposi- lion, so may a different interpretation be assicmed to the proposition itself. When the Wms are abst^Twe must reaa them in intension, and a proposition connecting such terms must denote the identity or non-identity of the qualities respectively denoted by the terms. Thus if we ^ay Equality = Identity of magnitude. 48 THE PRINCIPLES OF SCIENCE [chap. ni. 1^ f ( ti . ill. the assertion means that the circumstance of being equal exactly corresponds with the circumstance of being identical in magnitude. Similarly in Opacity = Incapability of transmitting light, the quality of being incapable of transmitting light is de- clared to be the same as the intended meaning of the word opacity. Wlien general names form the terms of a proposition we may apply a double interpretation. Thus Exogens = Dicotyledons means either that the qualities which belong to all exogens are the same as those which belong toall dicotyledons, orelse that every individual falling under one name falls equally under the other. Hence it may be said that there are two distinct fields of logical thought. We may argue either by the qualitative meaning of names or by the quantitative, that is, the extensive meaning. Every argument in- volving concrete pluml terms miglit be converted into one involving only abstract singular terms, and vice versd. But there are reasons for believing that the intensive or qualitative form of reasoning is the primaiy and fundamental one. It is sufficient to point out that the extensive meaning of a name is a changeable and fleeting thing, while the intensive meaning may nevertheless remain fixed. Very numerous additions have been lately made to the extensive meanings both of planet and element. Every iron steam-ship which is made or destroyed adds ta or subtracts from the extensive meaning of the name steam-ship, without necessarily affecting the intensive meaning. Stage coach means as much as ever in one way, but in extension the class is nearly extinct. Chinese railway, on the other hand, is a term represented only by a single instance ; in twenty years it may be the name of a large class. CHAPTER IV. DKDUCTIVK REASONING. The general principle of inference having been explained provided we have now before us the comparatively eas; task of tracing out the most common and impor Int fo'^s of deductive reasoning. The general problem of dedi^ tion IS a^ follows :-i^r.,;t one%r moreprM^^^^^ premuesto draw such otJur propositions iZu^es^^S ^^ be true wjun the premises are trvf. By deduction ZTyZi ) ^ gate and unfold the information contained 'nX^^^^^ and this we can do by one single vnle^For anyteZZ^r f ^ nru; in any proposition substitute the term whZ TaM ^ xnany premie to he identical unth it. To obtaTn cert^Q ies'SrrS^^^^ nf tI. IV- ! ^ ^'""^ '""^ "^® *^^ second and third Laws Jnatrect Deduction. In the present chapter however T shall confine my attention to those resX which cln^^ T^^^xCillK^^^ '^'"^^'^T "^^ ^"^^ of substitution system not on^v X "" • "^"^ ^°i^^^" ^^^^ «°^ harmonious system, not only the various moods of the ancient qvllnmor» biU a great number of equaUy important form of rj^^^^^^^ which had no recognised plL in the oldTo^c wT^^^^ apparatus 01 logical rules and mnemonic lines which 50 THE PRINCIPLES OF SCIENCE. [jUAI. IV.] DEDUCTIVE REASONING. 61 !li ' i i'( Immediate Inference, Probably the simplest of all forms of inference is that which has been called Immediate Inference, because it can be performed upon a single proposition. It consists in joining an adjective, or other qualifying clause of the same nature, to both sides of an identity, and asserting the equivalence of the terms thus produced. For instance, since Conductors of electricity = Non-electrics, it follows that Liquid conductors of electricity = Liquid non-electrics. If we suppose that Plants = Bodies decomposing carbonic acid, it follows that Microscopic plants = Microscopic bodies decotnposiug carbonic acid. In general terms, from the identity A = B we can infer tbe identity AC = BC. This is but a case of plain substitution; for by the rirst Law of Thought it must be admitted that AC = AC, and if, in the second side of this identity, we substitute for A its equivalent B, we obtain AC = BC. In like manner from the partial identity A = AB we may obtain AC = ABC by an exactly similar act of substitution ; and in every other case the rule will be found capable of verification by the principle of inference. The process when performed as here described will be quite free from the liability to error which I have shown ^ to exist in " Immediate Inference by added Determinants," as described by Dr. Thomson.^ ^ El&nuntary Lesions in Logic, p. 86. ' Outline of the L%tf« of Thought, § 87 Inference mth Tioo Simple Identities. One of the most common forms of inference, and one to which I shall especially direct attention, is practised with two simple identities. From the two statements that " London is the capital of England " and " London is the most populous city in the world," we instantaneously draw the conclusion that " The capital of England is the most populous city in the world." Similarly, from the identities Hydrogen = Substance of least density, Hydrogen = Substance of least atomic weight, we infer '^ Substance of least density = Substance of least atomic weight. The general form of the aigument is exhibited in the symbols B = A (,) B = 2 hence A=C. (3) We may describe the result by saying that terms identi- cal with the same term are identical with each other; and It IS impossible to overlook the analogy to the first axiom of Euclid that " things equal to the same thing are equal to each other." It has been very commonly supposed that this IS a fundamental principle of thought, incapable of reduction to anything simpler. But I entertain no doubt that this form of reasoning is only one case of the general rule of inference. We have two propositions, A = B and B = G, and we may for a moment consider the second one as affirming a truth concerning B, wliile the former one informs us that B is identical with A ; hence by substitu- tion we may affirm the same truth of A. It happens in this particular case that the truth affirmed is identity to G, and we might, if we preferred it, have considered the substitution as made by means of the second identity in tiie first. Having two identities we have a choice of the mode in which we will make the substitution, though thf result IS exactly the same in either case. Now compare the three following formulse (i) A = B = C, hence A = C (2) A = B - C, hence A - G (3) A '*' B -^ C, no infei-ence. « \ 52 THE PRINCIPLES OF SCIENCE. [chap 1 i I In the second formula we have an identity and a differ- ence, and we are able to infer a difference ; in the third we have two differences and are unable to make any inference at all. Because A and C both differ from B, we cannot tell whether they will or will not differ from each other, rhe flowers and leaves of a plant may both differ in colour from the earth in which the plant grows, and yet they may differ from each other ; in other cases the leaves and stem may both differ from the soil and yet agree with each other. Where we have difference only we can make no inference ; Avhere we have identity we can infer. This fact gives great countenance to my assertion that inference proceeds always through identity, but may be equally well effected in pro- positions asserting difference or identity. Deferring a more complete discussion of this point, I will only mention now that arguments from double identity occur very frequently, and are usually taken for granted, owing to their extreme simplicity. In regard to the equi- valence of words this form of inference must be constantly employed. If the ancient Greek 'voKko^ is our copper ^ then it must be the French cuivre, the German kup/er, the Latin cuprum^ because these are words, in one sense at least, equivalent to copper. Whenever we can give two defini- tions or expressions for the same term, the formula applies ; thus Senior defined wealth as " All those things, and those things only, which are transferable, are limited in supply, and are directly or indirectly productive of pleasure oi preventive of pain." Wealth is also equivalent to " things which have value in exchange ; " hence obviously, " things which have value in exchange = all those things, and those things only, which are transferable, «&;c." Two expressions for the same term are often given in the same sentence, and their equivalence implied. Thus Thomson and Tait say,^ ^The naturalist may be content to know matter as that which can be perceived by the senses, or as that which 3an be acted upon by or can exert force." I take this to mean — Matter = what can be perceived by the senses ; Matter =« what can be acted upon by or can exert force. I Trealvm on Naturtil PkUouophyt voL i. |>. l6l. IT,] DEDUCTIVE REASONING. 53 For the term ''matter" in either of these identities we may substitute its equivalent given in the other definition. Elsewhere they often employ sentences of the form exem- plified in the following:* "The integral curvature, or whole change of direction of an arc of a plane curve, is the angle through which the tangent has turned as we pass from one extremity to the other." This sentence is certainly of the form — The integral curvature = the whole change of direc- tion, &c. = the angle through which the tangent has turned, &c. Dis^'uised cases of the same kind of inference occur throughout all sciences, and a remarkable instance is found in algebraic geometry. Mathematicians readily show that every equation of the form y = mx -{■ c corresponds to or represents a straight line ; it is also easily proved that the same equation is equivalent to one of the general form Ac -I- By 4- C = o, and vice versd. Hence it follows that every equation of the form in question, that is to say, every equation of the first degree, corresponds to or represents a straight line.* In/errnce with a Simple and a Partial Identity. A form of reasoning somewhat different from that last considered consists in inference between a simple and a partial identity. If we have two propositions of the forms A = B, B = BC, we may then substitute for B in either proposition its equivalent in the other, getthig in both cases A = BC ; in this we may if we like make a second substitution for l>i getting A = AC. Thus, since « The Mont Blanc is the highest mountain in il-urope, and the Mont Blanc is deeply covered with snow ' we infer by an obvious substitution that "The highest mountain in Europe is deeply covered with snow." These propositions when rigorously stated faU into the forms above exhibited. This mode of inference is constantly employed when foi \ Tv!?u*"** ^ Natural Philosophy, vol. i. p. 6. lodiiuuter's Flaw dj-ordimU G&jwary, chap. ii. pp. ii— 14 • kHI 64 i: ii If l'*i I THE PRINCIPLES OF SCIENCE. [chap. a term we substitute its definition, or vice vend. The very purpose of a definition is to allow a single noun to be employed in place of a long descriptive phrase. Thus, when we say " A circle is a curve of the second degree," we may substitute a definition of the circle, getting " A curve, all points of which are at equal distances from one point, is a curve of the second degree." The real forms of the pro- positions here given are exactly those shown in the sym- bolic Statement, but in this and many other cases it will be sufficient to state them in ordinary elliptical language for sake of brevity. In scientific treatises a term and its definition are often both given in the same sentence, as in " The weight of a body in any given locality, or tho force with which the earth attracts it, is proportional to its mass." The conjunction or in this statement gives the force of equivalence to the parenthetic phrjise, so that tlie propositions really are Weight of a body = force with which the eartli attracts it Weight of a body = weight, &c. proportional to its mass. A slightly different case of inference consists in substitut- ing in a proposition of the form A = AB, a definition of the term B. Thus from A = AB and B =- C we get A = AC. For instance, we may say that " Metals are elements " and " Elements are incapable of decomposition." Metal = metal element. Element = what is incapable of decomposition. Hence Metal = metal incapable of decomposition. It is almost needless to point out that the form of these arguments does not sufTer any real modification if some Df the terms happen to be negative ; indeed in the last example " incapable of decomposition " may be treated as a negative term. Taking A = metal C = capable of decomposition B = element c = incapable of decomposition ; ihe propositions are of the forms A = AB B = c wnence, by substitution. A -- A£. IV.] DEDUCTIVE REASONING bh Infereiice of a Partial from Two Partial Identities. However common be the cases * of inference already noticed, there is a form occurring almost more frequentlv, and which deserves much attention, because it occupied'a prominent place in the ancient syllogistic system That system strangely overlooked all the kinds of argument we have as yet considered, and selected, as the type of all reasoning, one which employs two partial identities as premises. Thus from the propositions Sodium is a metil (i) Metals conduct electricity, (2) we may conclude that Sodium conducts electricity. (3) Taking A, B, C to represent the three terms respectively, the premises are of the forms A=AB (I) B ^ BC. (2) Now for B m (i) we can substitute its expression as given in (2), obtaining A = ABC, (3) or, in words, from Sodium =r sodium metil, (i\ Metal = metil conducting electricity, (2) we infer Sodium = sodium metal conducting electricity, (3 which, m the elliptical language of common life, becomes " Sodium conducts electricity." The above is a syllogism in the mood called Barbara ^ in the truly barbarous language of ancient logicians ; and the nret hgure of the syllogism contained Barbara and three other moods which were esteemed distinct forms of argu- ment But it is worthy of notice that, without any real Change in our form of inference, we readily include these three other moods under Barbara. The negative mood telarent wiU be represented by the example Neptune is a planet, (i) No planet has retrograde motion ; (2) iience Neptune has not retrograde motion. (3) Willi!! T^^T-^'"''' ""^ ^^'^ ^"'^ other technical terras of the old loric 56 THE PRINCIPLES OF SCIENCE. icnAP. If we put A for Neptune, B for planet, and C for " having retrograde motion," then by the corresponding negative term c, we denote "not having retrograde motion." The premises now fall into the forms A = AB (,) B = Be, (2) and by substitution for B, exactly as before, we obtain A = ABc (3) What is called in the old logic a particular conclusion may be deduced without any rcal variation in the symbols. Particular quantity is iodicated as before mentioned (p. 41), by joining to the term au indefinite adjective of quantity, such as sojtie, a part of, certain, &c., meaning that an unknown part of the term enters into the proposition as subject. Considerable doubt and ambiguity arise out of the question whether the part may not in some cases be the whole, and in the syllogism at least it must be under- stood in this sense.^ Now, if we take a letter to represent this indefinite part, we need make no change in our fornmlae to express the syllogisms Darii and Feria Con- uder the example — Some metals are of less density than water, (i) All bodies of less density than water will float upon the surface of water ; hence (2) Some metals will float upon the suiface of water. Let A = some metals, B = body of less density than water, C = floating on the surface of water then the propositions are evidently as before A = AB, B = BC; hence A = ABC, Thus the syllogism Darii does not really differ from^l(ar- bara. If the reader prefer it, we can readily employ a distinct symbol for the indefinite sign of quantity. Let P = some, Q = metal, B and C having the same meanings as before. Then the premises become (3) (I) (3' * SUmmtary Leuom in LogUf pp. 67, 79. ir] DEDUCTIVE REASONING. 57 PQ = PQB, (,) B = BC; (2) hence, by substitution, as before, PQ = PQBC. (3) Except that the formulas look a little more complicated there is no difference whatever. The mood Ferio is of exactly the same character as Darii or Barbara, except that it involves the use of a negative term. Take the example, Bodies which are equally elastic in all directions do not doubly refract light ; Some crystals are bodies equally elastic in all direc- tions; therefore, some crystals do not doubly refract light. Assigning the letters as follows : — A = some crystals, B = bodies equally elastic in all directions, C = doubly refracting light, c = not doubly refracting liglit. Our argument is of the same form as before, and may be concisely stated in one line, A = AB = ALc. If It IS preferred to put PQ for the indefinite same crystals we have PQ - PQB = PQBc. Ihe only diflerence is that the negative terra c takes the place of C in the mood Darii Ellipsis of Terms in. Partial Identities. The reader will probably have noticed that the conclu- sion which we obtain from premises is often more full than that drawn by the old Aristotelian processes. Thus from bodium IS a metal," and « Metals conduct electricity," we inferred (p. 55) that - Sodium = sodium, metal, conduct^ lu^. „ o^^^'^^*^>'>" whereas the old logic simply concludes that Sodium conducts electricity." SymboHcally, from A = AB, and B = BC, we get A = ABC, whereas the old logic gets at the most A = AC. It is therefore well to snow tliat without employing any other principles of A - 7nV^^'^" *^^'^^^ ''^^^'^^^y described, we may infer A - AO trom A = ABC, though we cannot infer the latter i \ I <J 68 THE PRINCIPLES OF SCIENCE. [chap more full and accurate result from the former. We may show this most simply as follows : — By the first Law of Thought it is evident that AA = AA; and if we have given the proposition A = ABC, we may substitute for both the A's in the second side of the above, obtaining AA = ABC . ABC. But from the property of logical symbols expressed in the Law of Simplicity (p. 33) some of the repeated lettera may be made to coalesce, and we have A = ABC . C. Substituting again for ABC its equivalent A, we obtain A = AC, tlie desired result. By a similar process of reasoning it may be shown that we can always drop out any term appearing in one member of a proposition, provided that we substitute for it the v/hole of the other member. This process was described in my first logical Essay,^ as Intrinsic Mimination, but it might perhaps be better entitled the Ellipsis of Terms. It enables us to get rid of needless terms by strict substitutive reasoning. Inference of a Simple from Two Partial Identities. Two terms may be connected together by two partial identities in yet another manner, and a case of inference then arises which k of the highest importance. In the two premises A = AB (i) B = AB (2) the second member of each is the same ; so that we can by obvious substitution obtain A = B. Thus, in plain geometry we readily prove that " Every equilateral triangle is also an equiangular triangle," and we can with equal ease prove that " Every equiangular triangle is an equilateral triangle.' Thence by substitution, as explained above, we pass to the simple identity. Equilateral triangle = equiangular triangle. ' Fure Logic, p. 19. «^l DEDUCTIVE REASONING. 59 We thus prove that one class of triangles is entirely identical with another class; that is to say, they differ only m our way of naming and regarding them. The great importance of this process of inference arises from the fact that the conclusion is more simple and general than either of the premises, and contains as much informa- tion as both of them put together. It is on this account constantly employed in inductive investigation, as wiU afterwards be more fully explained, and it is the natural mode by which we arrive at a conviction of the truth of simple identities as existing between classes of numerous objects. Inference of a Limited from Two Partial Identities. We have considered some arguments which are of the type treated by Aristotle in the first figure of the syllogism But there exist two other types of argument which employ a pair of partial identities. If our premises are as shown in these symbols, B = AB (X) B = CB, U we may substitute for B either by (i) in (2) or by (2) in (I), and by both modes we obtain the conclusion AB = CB, (.) a proposition of the kind which we have caUed a limited Identity (p. 42). Thus, for example, Potassium = potassium metal (i) Potassium = potassium capable of floating on hence ' ^^^ Potassium metal = potassium capable of float- Tk- • ^?,g^n water. /x Ihis IS really a syllogism of the mood Darapti in the tliird fagure, except that we obtain a conclusion ot' a more exact chamcter than the old syllogism gives. From the premises Po assium is a metal ^ and "Potassium floats on water," Aristotle would have inferred that "Some metals float on metik". .1 '''^''''^ "^^"^ "^^^^ ^h*<^ the "some T^lL .^'•*^>„t^^e^a«swer would certainly be "Metal which ^potassium " Hence Aristotle's conclusion simply leaves out some of the information afforded in the premises • it ;i eo THE PRINCIPLES OF SCIENCE. [CQAr. J \ even leaves us open to interpret the scmit metals in a wider sense than we are warranted in doing. From these distinct defects of the old syllogism the process of substitution is free, and the new process only incurs the possible objection of being tediously minute and accurate. Miscellaneous Forms of Deductive Inference. The more common forms of deductive reasoning having been exhibited and demonstrated on the principle of substitution, there still remain many, in fact an indefinite number, whicli may be explained with nearly equal ease. Such as involve the use of disjunctive propositions will be described in a later chapter, and several of the syllogistic moods which include negative terms will be more con- veniently treated after we have introduced the symbolic use of the second and third laws of thought. We sometimes meet with a chain of propositions which allow of repeated substitution, and form an argument called in the old logic a Sorites. Take, for instance, the premises Iron is a metal, (i) Metals are good conductors of electricity, (2) Good conductors of electricity are useful for telegraphic purposes. (3) It obviously follows that Iron is useful for telegraphic purposes. (4) Now if we take our letters thus, A = Iron, B = metal, C = good conductor of electricity, D = useful for telegraphic purposes, the premises will assume the forms A = AB, (I) B = BC, (2) C = CD. (3) For B in (i) we can substitute its equivalent in (2) obtaining, as before, A = ABC. Substituting for C in this intermediate result its equivalent as given in (3), we obtain the complete conclusion A = ABCD. (4) The full interpretation is that Iron is iron, m,etal, good conductor of electricity ^ usefvX for telegraphic purposes, which IT.] DEDUCTIVE REASONINO. 61 is abridged in common language by the ellipsis of the circumstances which are not of immediate importance. Instead of all the propositions being exactly of the same kind as in the last example, we may have a series of premises of various character ; for instance. Common salt is sodium chloride, (i) Sodium chloride crystallizes in a cubical form, (2) What crystallizes in a cubical form does not possess the power of double refraction : (7 it will foUow that • ^^ Common salt does not possess the power of double refraction. u) Taking our letter-terms thus, A = Common salt, B = Sodium chloride, C = Crystallizing in a cubical form, D = Possessing the power of double refraction, . we may state the premises in the forms A = B, (,) B = BC, I2) C = Cd, (3) Substituting by (3) in (2) and then by (2) as thus altered m (i) we obtain A = BCrf, (4) which is a more precise version of the common conclusion. We often meet with a series of propositions describing the qualities or circumstances of the one same thing, and we may combine them all into one proposition by the process of substitution. This case is, in fact, that which Or. Thomson has called "Immediate Inference by the sum of several predicates," and his example will serve my purpose well} He describes copper as "A metal— of a red colour— and disagreeable smell— and taste— all the preparations of which are poisonous— which is highly "lalleable— ductile— and tenacious— with a specific gravity of about 8.83." If we assign the letter A to copper, and the succeedmg letters of the alphabet in succession to the series of predicates, we have nine distinct statements, of the form A = AB (I) A = AC (2) A = AD (3) A = AK (9). we can readily combine these propositions into one by ' Ah OuiUnt of the Necessary Lam of Thought, Filth Ed. p. 161. ^ i'/ «s THE PRINCIPLES OP SCIENCE. [cHAlr. substituting for A in the second side of (i) its expression in (2). We thus get A = ABC, and by repeating the process over and over again we obviously get the single proposition A = ABCD . . JK. But Dr. Thomson is mistaken in supposing that we can obtain in this manner a definition of copper. Strictly speaking, the above proposition is only a description of copper, and all th« ordinary descriptions of substances in scientific works may be summed up in this form. Thus we may assert of the organic substances called Paraffins that they aie all saturated hydrocarbons, incapable of unitijig with other substances, produced by heating the alcoholic iodides with zinc, and so on. It may be shown that no amount of ordinary description can be equivalent to a de- finition of any substance. Fallacies, I have hitherto been engaged in showing that all the forms of reasoning of the old syllogistic logic, and an indefinite number of other forms in addition, may be readily and clearly explained on the single principle of ^^ substitution. It is now desirable to show that the same U principle will prevent us falling into fallacies. So long as we exactly observe the one rule of substitution of equivalents it will be impossible to commit a paralogism, that is to break any one of the elaboi-ate rules of the ancient system. The one new rule is thus proved to be as powerful as the six, eight, or more rules by wliich the cor- rectness of syllogistic reasoning was guarded. It was a fundamental rule, for instance, that two nega- tive premises could give no conclusion. If we take the propositions Granite is not a sedimentary rock, (l) Basalt is not a sedimentary rock, (2) we ought not to be able to draw any inference concerning the relation between granite and basalt. Taking our letter-terms thus : A = granite, B = sedimentary rock, C -= basalt, the premises may be expressed in the forms ^] DEDUCnVE REASONING. A - B, (1) C - B. (2) We have m this form two statements of difference; but the principle of inference can only work with a statement of agreement or identity (p. 63). Thus our rule gives us no power whatever of drawing any inference ; this is exactly in accordance with the fifth rule of the syllogism. It is to be remembered, indeed, that we clainf the power of always turning a negative proposition into an affirmative one (p. 45) ; and it might seem that the old rule agamst negative premises would thus be circumvented. Let us try. The premises (i) and (2) when affirmatively stated take the forms A = Aft (I) C = Cb. (2) The reader will find it impossible by the rule of substitu- tion to discover a relation between A and C. Three terms occur m the above premises, namely A, b, and C ; but they are so combmed that no term occurring in one has its exact equivalent stated in the other. No substitution can therefore be made, and the principle of the fifth rule of the syllogism holds true. Fallacy is impossible. It would be a mistake, however, to suppose that the mere occurrence of negative terms in both premises of a syllogism renders them incapable of yielding a conclusion. Ihe old rule informed us that from two negative premises DO conclusion could be drawn, but it is a fact that the rule m this bare form does not hold universaUy true • and I am not awai-e that any precise explanation has been Lnven of the conditions under which it is or is not imperative, l^onsider the following example : Whatever is not metallic is not capable of power- ful magnetic influence, d) Carbon is not metallic, )2) Therefore, carbon is not capable of powerful man- netic influence. °/^x r^rtr ^ff/^^ distinctly negative premises (i) and sfon r^f ^l 7^ ^'^^^ * perfectly valid negative conclu- sion (3). The syllogistic rule is actually falsified in its bare and general statement In this and many other cases we can convert the propositions into affirmative ones which will yield a conclusion by substitution without any difficulty il I i) u li) I ^ m i I! ill 1 Itl 64 THE PRINCIPLES OF SCIENCE. [cHAf To show this let A = carbon, B = metallic, C = capable of powerful magnetic influence. The premises readily take the fonns 6 = 6c, (f ; A = A6, (2) and substitution for h in (2) by means of (i) gives the conclusion . ^ A = Ahc. (3) Our principle of inference then includes the nile of negative premises whenever it is true, and discriminates correctly between the cases where it does and does not hold true. ^ rr j- The paralogism, anciently called the Fallacy of Undts- trihuted Middle, is also easily exhibited and infallibly avoided by our system. Let the premises bo Hydrogen is an element, (l J All metals are elements. (2) According to the syllogistic rules the middle term "element " is here undistributed, and no conclusion can be obtamed ; we cannot tell then whether hydrogen is or is not a metal Represent the terms as follows A = hydrogen, B = element, C = metal. The premises then become *^ A = AB, (i^ C = CB. (2) The reader will here, as in a former page (p. 62), find it impossible to make any substitution. The only term which occurs in both premises is B, but it is differently combined in the two premises. For B we must not substitute A, which is equivalent to AB, not to B. Nor must we confuse together CB and AB, which, though they contain one coin- mon letter, are different aggregate terms. The ride ot sub- stitution gives us no right to decompose combinations ; and if we adhere rigidly to the rule, that if two terms are stated to be equivalent we may substitute one for the other, we cannot commit the fallacy. It is apparent that the form of premises stated above is the same as that which wc obtained by translating two negative premises mto the affirmative form. iv.J DEDUCTIVE REASONING. The old fallacy, technically called the Illicit Process of the Major Term, is more easy to commit and more difficult to detect than any other breach of the syllogistic rules. In our system it could hardly occur. From the premises All planets are subject to gravity, (i) Fixed stars are not planets, (2) we might inadvertently but fallaciously infer that, " Fixed stars are not subject to gravity." To reduce the premises to symbolic form, let A = planet J^ = fixed star C = subject to gravity ; then we have the propositions A = AC (I) B = Ba. (2) The reader will try in vain to produce from these premises by legitimate substitution any relation between B and C ; he could not then commit the fallacy of asserting that B is not G. There remain two other kinds of paralogism, commonly known as the fallacy of Four Terms and the Illicit Process of the Minor Teim. They are so evidently impossible while we obey the rule of the substitution of equivalents, that it is not necessary to give any illustrations. When there are four distinct terms in two propositions a.s in A = B and C = U, tiiere cuuld evidently ije no opening for substitution. As to the Illicit Process of the Minor Terra it consists in a flagrant substitution for a term of another wider term which is not known to be. equivalent to it, and which is therefore not allowed by our rule to be •ubstituted for it CHAP, v.] DISJUNCTIVE PROPOSITIONS. €f t» CHAPTER V. i)iSJUNCTlVJi l>KOPOSITiON8. In the previous chapter I have exhibited various cases of deductive reasoning by the process oi substitution, avoid- ing the introduction of disjunctive propositions ; but we cannot long defer the consideration of this more complex class of identities General terms arise, as we liave seen (p. 24), from classifying or mentally uniting together all objects which agree in certain qualities, the value of this union consisting in the fact that the power of knowledge is multiplied thereby. In forming such classes or general notions, we overiook or abstract the points of difference which exist between the objects joined together, and fix our attention only on the points of agreement But every process of thought may be said to have its inverse process, wliich consists in undoing the effects of the direct process. Just as division undoes multiplication, and evolution un- does involution, so we must have a process which undoes generalization, or the operation of forming general notions. This inverse process will consist in distinguishing the separate objects or minor classes which are the constituent parts of any wider class. If we mentally unite together certain objects visible in the sky and call tliem planets, we shall afterwards need to distinguish the contents of this general notion, which we do in the disjunctive proposi- tion — A planet is either Mercury or Venus or the Earth or or Neptune. Having formed the very wide class " vertebrate animal," we may specify its subordinate classes thus : — " A verte- brate animal is either a mammal, bird, reptile, or fish." Nor is there any limit to the number of possible altema tives. "An exogenous plant is either a ranunculus, a poppy, a cnicifer, a rose, or it belongs to some one of -the other seventy natui-al orders of exogens at present recog- nized by botanists." A cathedral church in England must be either that of London, Canterbury, Winchester, Salis- bury, Manchester, or of one of about twenty-four cities possessing such churches. And if we were to attempt to specify the meaning of the term " star," we should require to enumerate as alternatives, not only the many thousands of stars recorded in catalogues, but the many millions un- named. Whenever we thus distinguish the parts of a general notion we employ a disjunctive proposition, in at least one ^^ side of which are several alternatives joined by the so- called disjunctive conjunction or, a contracted form of other. There must be some relation between the parts thus con- nected in one proposition ; we may call it the disjwnctive or alternative relation, and we must carefully inquire into its nature. This relation is that of ignorance and doubt, giving rise to choice. Whenever we classify and abstract we must open the way to such uncertainty. By fixing our attention on certain attributes to the exclusion of others we necessarily leave it doubtful what those other attributes are. The term " molar tooth " bears upon the face of it that It is a part of the wider term " tooth." But if we meet with the simple term " tooth " there is nothing to in- dicate whether it is an incisor, a canine, or a molar tooth. Ihis doubt, however, may be resolved by further informa- tion, and we have to consider what are the appropriate logical processes for treating disjunctive propositions in connection with other propositions disjunctive or otherwise. Expression of tlie Alternative Relation. In order to represent disjunctive propositions with con- venience we require a sign of the alternative relation, equivalent to one meaning at least of the little conjunc- tion or so frequently used in common language. I pro- pose to use for this purpose the symbol .|. . In my first logical essay I followed the practice of Boole and adopted F 2 iS i j t: i ii d8 THE PRINCIPLES OF SCIENCE. [cBAr. the sign +; but this sign should not be employed unless there exists exact analogy between mathematical addition and logical alternation. We shall find that the analogy is im- perfect, and that there is such profound difference between logical and mathematical terms as should prevent our uniting them by the same symbol. Accordingly I have chosen a sign •!• , which seems aptly to suggest whatever degree of analogy may exist without implying more. The exact meaning of the symbol we will now proceed to investigate. Nature of the Alter native Relation. Before treating disjunctive propositions it is indispens- able to decide whether the alternatives must be considered exclusive or unexclusive. By exclusive aitemativcs we mean those which cannot contain the same things. If we say " Arches are circular or pointed," it is certainly to be understood that the same arch cannot be described as both circular and pointed. Many examples, on the other hand, 3an readily be suggested in which two or more alteraatives may hold true of the same object. Thus Luminous bodies are self-luminous or luminous by reflection. It is undoubtedly possible, by the laws of optics, that the 3ame surface may at one and the same moment give ofl* light of its own and reflect light from other bodies. We speak familiarly of cka/or dumb persons, knowing that the majority of those who are deaf from birth are also dumb. There can be no doubt that in a great many cases, perhaps the greater number of cases, alternatives are exclusive as a matter of fact. Any one number is incompatible with any other ; one point of time or place is exclusive of all others. Roger Bacon died either in 1284 or 1292 ; it is certain ^hat he could not die in both years. Henry Fielding was born either in Dublin or Somersetshire; he could not be born in both places. There is so much more precision and clearness in the use ot exclusive alternatives that we ought doubtless to select them when possible. Old works on logic accordingly wntained a rule directing that the Membra divideniia, the '.J DISJUNCTIVE PROPOSITIONS. 09 parts of a division or the constituent species of a genus, should be exclusive of each other. It is no doubt owing to the great prevalence and con- venience of exclusive divisions that the majority of logi- cians have held it necessary to make every alternative in a disjunctive proposition exclusive of every other one Aquinas considered that when this was not the case the proposition was actually false, and Kant adopted the same opinion.* A multitude of statements to the same eHect might readily be quoted, and if £he question were to be determined by the weight of historical evidence It would certainly go against my view. Among recent logicians Hamilton, as well as Boole, took the exclusive V^^ i^".*^r *^®^ *^® authorities to the opposite effect. A\hately, Mausel, and J. S. Mill have all pointed out that we may often treat alternatives as Compossible, or true at the same time. Whately gives us an example,^ " Virtue tends to procure us either the esteem of mankind, or the favour of God," and he adds—" Here both members are true, and consequently from one being aftirmed we are not authorized to deny the other. Of course we are left to conjecture in each case, from the context, whether it is meant to be implied that the members are or are nor exclusive." Mansel says,^ " JVe mai/ happen to know that two alternatives cannot be true together, so that the athrmation of the second necessitates the denial of the hrst ; but this, as I^thius observes, is a material, not a formal consequence." Mill has also pointed out the absurdities which would arise from always interpreting alternatives as exclusive. « If we assert," he says,* " that a man who has acted in some particular way must be cither a knave or a fool, we by no means assert, or intend to assert, that he cannot be both." Again, "to make an entirely unselfish use of despotic power, a man must be either a saint or a philosopher. Does the dis- junctive premise necessarily imply, or must it be construed as supposing, that the same person cannot be both a I MansePs Aldrich, p. 103, and ProUgomma Logica, p. 221. , KUmenU of Logic, Book II. chap. iv. sect. 4. ^ Aldricb, Artis Logica: Budimenta, p. 104. t^ramiiiaium of Sir W. Hamilton's Philosophy, pp. 452-454. If i J , - t \ \ i x I 'I ,NM f 1^ |. n THE PRINCIPLES OF SCIENCE. [chap. saint and a philosopher ? Such a construction would be ridiculous." I discuss this subject fully because it is really the point which separates my logical system from that of Boole. In his Laws of Thoiujht (p. 32) he expressly says, ** In strictness, tlie words * and,' * or,' interposed Ijetween the terms descriptive of two or more classes of objects, imply that those classes are quite distinct, so that no member of one is found in another." This I altogether dispute. In the ordinary use of these conjunctions we do not join distinct terms only ; and when terms so joined do prove to be logically distinct, it is by virtue of a tacit •premise, something in the meaning of the names and our knowledge of them, which teaches us that they are distinct. If our knowledge of the meanings of the words joined is defective it will often be impossible to decide whether tenns joined by conjunctions are exclusive or not. In the sentence " Repentance is not a single act, but a habit or virtue," it cannot be implied that a virtue is not a habit ; by Aristotle's definition it is. Milton has the expression in one of his sonnets, " Unstain'd by gold or fee," where it is obvious that if the fee is not always gold, the gold is meant to be a fee or bribe. Tennyson has the expression " wreath or anadem." Most readers would be quite uncertain whether a wreath may be an anadem, or an anadem a wreath, or whether they are quite distinct or quite the same. From Darwin's Origin of Species, I take the expression, "When we see any part or organ developed in a remarkable degree or manner." In this, or is us«d twice, and neither time exclusively. For if part and organ are not synonymous, at any rate an organ is a part. And it is obvious that a part may be ileveloped at the same time both in an extraordinary degree and an extraordinary manner, although such cases may be com- paratively rare. From a careful examination of ordinary writings, it will tlms be found that the meanings of terms joined by "and," " or " vary from absolute identity up to absolute contrariety. There is no logical condition of distinctness at all, and when we do choose exclusive alternatives, it is because our subject demands it The matter, not the form of an fj DISJUNCTIVE PHOPOSITIONS. 71 expression, points out whether terms are exclusive or not.' In bills, policies, and other kinds of legal documents, it is sometimes necessary to express very distinctly that alternatives are not exclusive. The form — is then or used, and, as Mr. J. J. Murphy has remarked, this form coincides exactly in meaning with the symbol .|. . In the first edition of this work (vol. i., p. 81), I took the disjunctive proposition " Matter is solid, or liquid, or gaseous," and treated it as an instance of exclusive altern- atives, remarking that the same portion of matter cannot be at once solid and liquid, property speaking, and that still less can we suppose it to be solid and gaseous, or solid, liquid, and gaseous all at the same time. But the experiments of Professor Andrews show that, under certain conditions of temperature and pressure, there is no abrupt change from the liquid to the gaseous state. The same substance may be m such a state as to be indiflerently described as liquid and gaseous. In many cases, too, the transition from solid to liquid is gradual, so that the properties of solidity are at least partially joined with those of liquidity. The proposition then, instead of being an instance of exclusive alternatives, «eems to afford an excellent instance to the opposite effect. When such doubts can arise, it is evidently impossible to treat alternatives as absolutely exclusive by the logical nature of the relation. It becomes purely a question of the matter of the proposition. The question, as we shall afterwards see more fully, is one of the greatest theoretical importance, because it concerns the true distinction between the sciences of Logic and Mathematics. It is the foundation of number that every unit shall be distinct from every other unit ; but Boole imported the conditions of number into the science of Logic, and produced a system which, though wonderful in its results, was not a system of logic at all. Laws of tlie Diy'unctive Relation. In considering the combination or synthesis of terms (P- 30), we found that certain laws, those of Simplicity > Pwc Logic, pp 76, 77. I mV^ "V ■,' J ' < I* il\ t( >.< o 72 THE PRINCIPLES OF SCIENCE. [chap. and Commutativeness, must be observed. In uniting terms by the disjunctive symbol we shall find that the same or closely similar laws hold true. The all ^natives of either member of a disjunctive proposition are certainly commutative. Just as we cannot properly distinguish between rich and rare gems and rare and rich gems, so we must consider as identical the expression rich or rare gems, and rare or rich gems. In our symbolic language we may say A + B = 15 + A. The order of statement, in short, has no effect upon the meaning of an aggregate of alternatives, so that the Law of Commutativeness holds true of the disjunctive symbol. As we have admitted the possibility of joining as alter- natives t^rms which are not really different, the question arises. How shall we treat two or more alternatives when they are clearly shown to be the same? If we have it asserted that P is Q or R, and it is afterwanls proved that Q is but another name for R, the result is that P is either K or R How shall we interpret such a statement ? What would be the meaning, for instance, of " wreath or anadem " if, on referring to a dictionary, we found anadem described as a wreath ? I take it to be self-evident that the meaning would then l)ecome simply "wreath." Acconlingly we may affinn the general law A + A = A, Any number of identical alternatives may always be reduced to, and are logically equivalent to, any one of those alternatives. This is a law which distinguishes mathematical terms from logical terms, because it obviously does not apply to the former. I propose to call it the Law of Unity, because it must really be involved in any definition of a mathematical unit This law is closely analogous to the Law of Simplicity, AA = A ; and the nature of the connection is worthy of attention. Few or no logicians except De Morgan have adequately noticed the close relation between combined and disjunctive terms, namely, that every disjunctive term is the negative of a corresponding combined term, and vice versd. Consider the term Malleable dense metal V.) DISJUNCTIVE PROPOSITIONS. 78 How shall we describe the class of things which are not malleable-dense-metals ? Whatever is included under that terni must have all the qualities of malleability, denseness and metalhcity. Wherever any one or more of the qualities IS wanting, the combined term will not apply. Hence the negative of the whole term is Not-malleable or not-dense or not-metallic. In the above the conjunction or must clearly be inter- preted as unexclusive; for there may readily be objects which arc both not-malleable, and not-dense, and ])ei-liaps not-metaUic at the same time. If in fact we were required to use or m a strictly exclusive manner, it would be requisite to specify seven distinct alternatives in order to describe the negative of a combination of three terms. I he negatives of four or five terms would consist of fifteen or thirty-one alternatives. This consideration alone is sufhcient to prove that the meaning of or cannot be always exclusive in common language. Expressed symbolically, we may say that the negative ABC is not-A or not-B or not-C ; that is, a I- b f c. Reciprocally the negative of P + Q I- R Every disjunctive term, then, is the negative of a combined term, and mce versd. Apply this result to the combined term AAA, and its negative is a •{• a -j- a. Since AAA is by the Law of Simplicity equivalent to A so a jr a J- a must be equivalent to a, and the Law of the^o'th ^^^^' ^^^ ^*^ ^'^"^ necessarily presupposes Symbolic expression of the Law of Ditality. We naay now employ our symbol of alternation to express in a clear and formal manner the third Funda- mental Law of Thought, which I have called the Law ot Duality (p. 6). Taking A to represent any class or i 74 H THE PRINCIPLES OF SCIENCE. [caAP. r.] DISJUNCTIVE PROPOSITIONS. 76 object or quality, and B any otlier class, object or quality, we may always assert that A either agrees with B, or does not agree. Thus we may say A = AB .|. Ab. This is a formula which will henceforth be constantly employed, and it lies at tlie basis of reasoning. The reader may perliaps wish to know why A is inserted in both alternatives of the second member of the identity, and why the law is not stated in the form A = B .|. b. But if he will consider the contents of the last section (p. 73), he will see that tlie latter expression cannot be correct, otherwise no term could have a corresponding negative term. For the negative of B .|. 6 is 6B, or a self- contradictory term ; thus if A were identical with B j. b its negative a would be non-existent. To say the least, this i-esult would in most cases be an absurd one, and I see much reason to think that in a strictly logical point ol view it would always be absurd. In all probability we ought to assume as a fundamental logical axiom that every iemi has its negative in thought. We cannot think at all without separating what we think about from other things, and these things necessarily form the negative notion.' It follows that any proposition of the form A = B J- 6 is just as self-contradictory as one of the form A = hb. It is convenient to recapitulate in this place the thru* Laws of Thought in their symbolic form, thus Law of Identity A = A. Law ol' (Juuirnuiutiou Au -= o. Law of Duality A = AB •!• Ab. Various Foiins of the Disjunctive Proposition. Disjunctive propositions may occur in a great variety of forms, of which the old logicians took insufficient notice. There may be any number of alternatives, each of which may be a combination of any number of simple terms. A proposition, again, may be disjunctive in one or both members. The proposition * Pure LogiCy p. 65. See also the criticism of this point by De Morgan in the Athenaum^ No. 1892, 30th January, 1864 ; p. 155. Solids or liquids or gases are electrics or conductors of electricity is an example of the doubly disjunctive form. The mean- mg of such a proposition is that whatever falls under any one or more alternatives on one side must fall under one or more alternatives on the other side. From what has been said before, it is apparent that the proposition A.|.B=C.|.D "^ will correspond to each member of the latter being the negative of a member ot the former proposition. As an instance of a complex disjunctive proposition I naay give Senior's definition of wealth, which; briefly stated, amounts to the proposition « Wealth is what is transferable, limited in supply, and either productive of pleasure or preventive of pain." * Let A = wealth B = transferable C = limited in supply D = productive of pleasure »,. , ^ E = preventive of pain. ihe definition takes the form K . r . A = BC(D.|.E); but If we develop the alternatives by a method to be afterwards more fully considered, it becomes A = BCDE .|. BCDe .|. BCrfE. foutiS TdJ^M ^^ ^ '™ .^^^ ^'^"^P^^^ proposition is thul, '^^ '^'"" ^^'^ ^^"^'^ ""^ *^^ ^'P^^^^^ ^ succession, ' A = he B = rich C ■» absolutely mad D « weakness itself E = subjected to bad advjce « ^It' ^^^'^^P- '^- ^'l^^' P^re Logic, p. 69. ^un ih. Syllogum, No. ,11. p. ,2. Camb. Phil, l^'n^ Vol. , ■ \i i( :■ i:^ 76 THE PRINCIPLES OP SCIENCE. [cHAfk F = subjected to most unfavourable circumstances, the proposition will take the form A = AB{C I- D (E I- F)}, and if we develop the alternatives, expressing some of the diflerent cases which may happen, we obtain A = ABC I- ABcDEF I- ABcDE/l ABcBcY, The above gives the strict logical interpretation of the sentence, and the first alternative ABC is capable of de- velopment into eight cases, according as 1), E and F are or are not present. Although from our knowledge of the matter, we may infer that weakness of character cannot be asserted of a person absolutely mad, there is no explicit statement to this eflect. Inference by Disjunctive Propositions. Before we can make a free use of disjunctive proposi- tions in the processes of inference we must consider how disjunctive terms can be combined together or with simple terms. In the first place, to combine a simple term with a disjunctive one, we must combine it with every alternative of the disjunctive term. A vegetable, for instance, is either a herb, a shrub, or a tree. Hence an exogenous vegetable is either an exogenous herb, or an exogenous shrub, or an exogenous tree. Symbolically stated, this process of combination is as follows, A(B.|.C) = ABlAC. Secondly, to combine two disjunctive terms with each other, combine each alternative of one with each alterna- tive of the other. Since flowering plants are either exogens or endogeus, and are at the same time either herbs, shrubs or trees, it follows that there are altogether six alternatives — namely, exogenous herbs, exogenous shrubs, exogenous trees, endogenous herbs, endogenous shrubs, endogenous trees. This process of combination is shown in the general form (A .|. B) (C .|. D .|. E) = AC I- AD .|. AE .| BC I- BD + BE It is hardly necessary to point out that, however numerous the terms combined, or the alternatives in those terms, we may effect the combination, provided each alter- native is combined with each alternative of the other terms, as in the algebraic process of multiplication. M DISJUNCTIVE PROPOSITIONS. 77 Some processes of deduction may be at once exhibited. We may always, for mstance, unite the same qualifyinff term to each side of an identity even though one or both members of the identity be disjunctive. Thus let A = B .|. C. Now it is self-evident that AD = AD. AD = BD + CD. Since a gaseous element is either hydrogen or owcen or nitiogen, or chlorine, or fluorine." it follows il.at "a free Saseous element ,s either free hydrogen, or free oxygen or free nitrogen, or free chlorine, or frJe fluorine " This process of combination will lead to most useful in- ^^^Tf^ ^* qualifying adjective combined with both sides of the proposition is a negative of one or more alter- natives. Since chlorine is a coloured gas, we may infer hat "a colourless gaseous element is either (colourless hydrogen oxygen, nitrogen, or fluorine." The altematrvi chlonne disappears because colourless chlorine does not exist Again, since "a tooth is eitl^r an incisor, canine b cuspid, or molar. • it follows that -''^ not-incisor l^rS either canme. bicuspid, or molar." The geneml rule is that from the denial of any of the altemativl the afflrmaUon of the remainder can be inferred. Now this result c earfy S7-Xr '"^^ "' ^"^"*"««" = '- 'f - >^- evilrt iSty " '""^^^^''^^'^^^ «- -'^'^ of the self- A6 = Ab, we obtain AJ = AB6 .|. A5C I- A JD • and as the first of the three alternatives is self-contra »,. AJ = AJC .|. A6D. the"'lZ,rf^'" «''^r '?"'"*^"* '^"'' «^Pl«'"« "'"t mood of tivfsv^l" Urn %'l?r' "^"^^""y ''^'''- *»t th" Disjunc- tive byUogism of the mood ponendo tollens. which affirms ■ 'i I ). .•/' 78 THE PRINCIPLES OF SCIENCE. [chap. ^.J DISJUNCTIVE PROPOSITIONS. 7» one alternative, and thence infers the denial of the rest, cannot be held true in this system. If I say, indeed, that Water is either salt or fresh water, it seems evident that " water which is salt is not fresh." But this inference really proceeds from our knowledge that water cannot be at once salt and fresh. This inconsistency of the alternatives, as I have fully shown, will not always hold. Thus, if I say Gems are either rare stones or beautiful stones, (i) it will obviously not follow that A rare gem is not a beautiful stone, (2) nor that A beautiful gem is not a rare stone. (3) Our symbolic method gives only true conclusions ; for if we take A = gem B = rare stone C = beautiful stone, the proposition (i) is of the form A = B .|. C hence AB = B I- BC and .^^ = BC.|.C; but these inferences are not equivalent to the false ones (2) and (3). We can readily represent disjunctive reasoning by the modus fonendo tollens, when it is valid, by expressing the inconsistency of the alternatives explicitly. Thus if we resort to our instance of Water is either salt or fresh, and take A = Water B = salt C = fresh, then the premise is apparently of the form A = ABl-AC; but in reality there is an unexpressed condition that " what is salt is not fresh," from which follows, by a process of inference to be afterwards described, that " what is fresh is not salt." We have then, in letter-terms, the two pro- positions B = B(j C = JC. If we substitute these descriptions in the original pre position, we obtain A = ABc .|. AhG ; uniting B to each side we infer AB = ABc .|. ABbG or AB = ABc ; that is, Water which is salt is water salt and not fresh. I should weary the reader if I attempted to illustrate the multitude of forms which disjunctive reasoning may take; and as in the next chapter we shall be constantly treating the subject, I must here restrict myself to a single instance. A very common process of reasoning consists in the determination of the name of a thing by the successive exclusion of alternatives, a process called by the old name abscissto mfimti. Take the case : Red-coloured metal is either copper or gold (i) Copper is dissolved by nitric acid (2) . This specimen is red-coloured metal (3) This specimen is not dissolved bv nitric acid (4) Therefore, this specimen consists" of gold f c) Let us assign the letter-symboU thus— A = this specimen D = gold B = red-coloured metal E = dissolved by nitric acid C = copper Assuming that the alternatives copper or (jold are intended to be exclusive, as just explained in the case of S ^^''^'' *^® P"^""^'^' ^^3^ ^ «^ted in the B = BCrf.|.BcD (i\ G = CE U 1 : 1? (3) Substituting for C in (i) by means of (2) we get ^^^ ^ ^ B = BCrfE -h Bel) i^rom (3) and (4) we may infer likewise A = AR; ^Jfit'foKs'har'^""^ '" ^ '"^ equivalent. just ^ A = ABGdEe I- ABcBe Uie first of the alternatives being contradictory the result A = AlicDe ' Hi 80 THK PRINCIPLES Ol! SCIENCE. [chap. v. L' 'Hj|;t which contains a full description of " this specimen " as furnished in the premises, but by ellipsis asserts that it is gold. It will be observed that in the symbolic expression (l) I have explicitly stated what is certainly implied, that copper is not gold, and gold not copper, without which condition the inference would not hold good. CHAPTER \X Tm INDIRKCT METHOD OF INFKRBNOE. The forms of deductive reasoning as yet considered, are mostly cases of Direct Deduction as distinguished from those which we are now about to treat. The method of Indirect Deduction may be described as that which points out what a thing is. by showing that it cannot be anything el3e. We can define a certain space upon a map, either by colouring that space, or by colouring all except the space ; the first, mode is positive, the second negative. The difference, it will be readily seen, is exactly analogous to that between the direct and indirect modes of proof in geometry. Euclid often shows that two lines are equal by showing that they cannot be unequal, and the proof rests jipon the known number of alternatives, greater, equal or loss, which (ire alone conceivable. In other cases, as for nisUmce m the seventh proposition of the first book, he Shows that two lines must meet in a particular point, by showing that they cannot meet elsewhere. In logic we can always define with certainty the utmost number of «altematives which are conceivable. The Law quality (pp. 6, 74) enables us always to assert that any ?hc ?" «^,circumstance whatsoever is either present or X'. Whatever may be the meaning of the terms A •urn 15 It IS certainly true that A = AB.|.A5 B = AB.|.aB. Teh Wo • M '""^^'^'^ ^^ ^^'^ P^^^l^"^> ^"^ ^J^ich e^ve «"tii invariable and necessary conditions of all thought, o ■ I \ V' 'I THE PRINCIPLES OF SCIENCE. [CHAF that they need not be specially laid down. The Law of Contradiction is a further condition of all thought and of all logical symbols; it enables, and in fact obliges, us to reject from further consideration all terms which imply the presence and absence of the same quality. Now, when- ever we bring both these Laws of Thought into explicit action by the method of substitution, we employ the Indirect Method of Inference. It will be found that we can treat not ouly those arguments already exhibited according to the direct method, but we can include an infinite multitude of other arguments which are incapable of solution by any other means. Some philosophers, especially those of France, have held that the Indirect Method of Proof has a certain inferiority to the direct method, which should prevent our using it except when obliged. But there are many truths which we can prove only indirectly. We can prove that a number is a prime only by the purely indirect method of showing that it is not any of the numbers which^ have divisors, and the remarkable process known as Eratos- thenes* Sieve is the only mode by which we can select the prime numbers.^ It bears a strong analogy to the indirect method here to be described. We can prove that the side and diameter of a square are incommensurable, but only in the negative or indirect manner, by showing tliat the con- trary supposition inevitably leads to contradiction.* Many other demonstrations in various branches of the mathe- matical sciences proceed upon a like method. Now, if there is only one important truth which must be, and can only be, proved indirectly, we may say that the process is a necessary and sufficient one, and the question of its com- parative excellence or usefulness is not worth discussion. As a matter of fact I believe that nearly half our logical conclusions rest upon its employment. ' SeeHorsley, Philosophical Transactioni, 1772 ; vol. Ixii. p. 327. Montucla, Histoire dts Matheviatuiwa, vol. i. p. 2^9. renny Cydopadicty article '* Eratosthenes." « Euclid, Book X. Prop. 117. ^ij THE lyPIREC TMETHOD OF INFERENCE. 83 Simple IllustrcUions. J\Sn%t^ 'l^ P""""? ''"'J '^"l'« of this method we has had th^lewt £cal ^i "•'*'"?«• ^"^ P^"^" ^»'<' dmw from tie abo^;^*v"^' " *^*'* t'"'* ^« <^^ one. namely, P^'P'^'tion an apparently different o^sldeZMf Sro^thr'^r ^^ ^"•«-'' have purely self-evident and neithTr '^^ P~P««'«on« ^ be analysis, a creat manv L^ ne^dmg nor aUowing while tekchfnrio^cLa^fiP!r"''u \^''^^ '^^'^^^ close connecti'on ^^^ ttrn'^lSelfaTr *^^ complete system of InrnV ^iii V ."Y^^^^e tJiat a true and this VeL. which Z^l cX r '^'■ ""^^^^ °f rem^; the full procesTisl^folwJ''^'"''^^ ^'«- lirstly. by the Uw of Duality we know that If .> i^°K'f""^^^^ Metal or Not-metal LL^ntid^tr,: s^ '' '' y *^«^-'"- - is an ekment and a n^L'^leS'X^ ''* ^"^ ^'''"g to the Law of ContradS ^ L-' "* «PPo«'tion other alteraative then 1 1 .V /According to the only metal. ' ^''' *''® "ot-elenient must be a not To represent this process of inference avmV^i;. ii take tlie premise in the form *""'® symbolicaUy we tTrdtria""* "' "" ^'^ ''' ^-"-'y *e term noS is ^'^tS):SS^^'^^ ^^ -^escriptio^a, O o2 r ^i 84 THE PRINCIPLES OF SCIENCE. [chap. fi.] THE INDIRECT METHOD OF INFERENCE. i(||;»' '\ I Hence it results that h is either nothing at all, or it is db; and the conclusion is As it will often be necessary to refer to a conclusion of this kind I shall call it, as is usual, the Cmtraposthve Proposition of the original. The reader need hardly be cautioned to observe that from all A's are B's it does not follow that all not-A's are not-B's. For by the Law of Duality we have and it will not be found possible to make any substitution in this by our original prendse A = AB. It still remains doubtful, therefore, whether not-metal is element or not- element. . . . , The proof of the Contrapositive Proposition given above is exactly the same as that which Euclid applies in the case of geometrical notions. De Morgan describes Euclid s process as follows^ :— " From every not-B is not- A he pro- duces Every A is B, thus : If it be possible, let this A be not-B, but every not-B is not-A, therefore this A is not-A, which is absurd : whence every A is B." Now Dc Morgan thinks that this proof is entirely needless, because common logic gives the inference without the use of any geo- metrical reasoning. I conceive however that logic gives the inference only by an indirect process. De Morgan claims " to see identity in Every A is B and every not-B is not-A, by a process of thought prior to syllogism. Whether prior to syllogism or not, I claim that it is not prior to the laws of thought and the process of substitutive inference, by which it may be undoubtedly demonstrated. Employmmt of the Contrapositive Proposition, We can frequently employ the contrapositive form of a proposition by the method of substitution ; and certain moods of the ancient syllogism, which we have hitherto passed over, may thus be satisfactorily comprehended in our system. Take for instance the following syUogism in the mood Camestres : — ' PkUosophicnl Afajfttti/n}, Dec. 1 852 ; p. 437. " Whales are not true fish ; for they do not respire water, whereas true fish do respire water." Let us take A = whale B = true fish C = respiring water Tlie premises are of the forms A = Ac . ,\ B = BC I J Now, by the process of contraposition we obtain from the second premise and we can substitute this expression for c in Ci) ob- taining ^ ^' A = Ahc or "Whales are not true fish, not respiring water" The mood Cesare does not really differ'' from Camestres except in the order of the premises, and it could be ex- hibited in an exactly similar manner. The m(K)d Baroko gave much trouble to the old logicians who could not reduce it to the first figure in the same manner as the other moods, and were obliged to invent specially for it and for Bokardo, a method of Indirect Reduction closely analogous to the indirect proof of Euclid. Now these moods require no exceptional treatment in this system. Let us take as an instance of Baroko, the areu ment ^ AU heated solids give continuous spectra (i) Some nebula do not give continuous spectra (2) Therefore, some nebulae are not heated solids (^) Treating the little word some as an indeterminate adiec- tive of selection to which we assign a symbol like any other adjective, let ^ A = some B = nebulsB C = giving continuous specti-a rp, ^ = heated solids A he premises then become D = DC (I) XT . AB = ABc (2) tl^I ^^'^ ^'^ "^^ ""^^"^ ^y *^^« ^°^^«^fc method the cou. trapositive proposition » / I i; f ^'i > ( THE PRINCIPLES OF SCIENCE. [chap. c = cd and if we substitute this expression for c in (2) we have AB = ABcd the full meaning of which is that " some nebulae do not give continuous spectra and are not heated solids." We ini^lit similarly apply the contrapositive in many other instances. Take the argument, " All fixed stars are self-luminous ; but some of the heavenly bodies are not self-luminous, and are therefore not fixed stars." Taking our terms A = fixed stars B = self-luminous C = some D = heavenly bodies we have the premises A = AB, (i) CD = bCD (2) Now from (i) we can draw the contrapositive & = aJ and substituting this expression for h in (2) we obtain CD = abCD which expresses the conclusion of the ai-gument that some heavenly bodies are not fixed stars. Contrapositive of a Simple Identity, The reader should carefully note that when we apply the process of Indirect Inference to a simple identity of the form A = B we may obtain further results. If we wish to know what is the term not-B, we have as before, by the I^aw of Duality, h = Ah •!• ah and substituting for A we obtain h =^W) \ah = ah. But we may now also draw a second contrapositive ; for we have a = aB •!• ah, and substituting for B its equivalent A we have a = a A \ah == ah. Hence from the single identity A = B we can draw the two propositions ru] THE INDIRECT METHOD OF INFERENCE. 87 a^ ah b = ab, and observing that these propositions have a common term ab we can make a new substitution, getting a = 5. This result is in strict accordance with the fundamental principles of inference, and it may be a question whether it IS not a self-evident result, independent of the steps of deduction by which we have reached it For where two classes are coincident like A and B, whatever is true of the one is true of the other ; what is excluded from the one must be excluded from the other similarly. Now as a bears to A exactly the same relation that h bears to B the identity of either pair follows from the identity of the other pair. In every identity, equality, or simUarity, we may argue from the negative of the one side to the nec^a- tive of the other. Thus at ordinary temperatures ^^ Mercury = liquid-metal, hence obviously • Not-mercury = not liquid-metal ; or since Sirius = brightest fixed star, it follows that whatever star is not the brightest is not fc>irius, and vice versd. Every correct dcfniition is of the lorm A = B, and may often require to be applied in the eqmvalent negative form. Let us take as an illustration of the mode of usin^r this result the argument following : ° Vowels are letters which can be sounded alone, (i) The letter w cannot be sounded alone ; ' (2) Therefore the letter lo is not a vowel. (3) Here we have a definition (i), and a comparison of a thing with that definition (2), leading to exclusion of the tiling from the class defined. Taking the terms A = vowel, B = letter which can be sounded alone, C = letter w, the premises are plainly of the forms A=B, (,j C = 6C. (2) I M I I H '1 i/ " ' M 88 THE PRINCIPLES OF SCIENCE. [OUAF. Now by the Indirect method we obtain from (i) the Contrapositive 6 = rt. and inserting in (2) the equivalent for 6 wo have C = aC, (3) IT " the letter w is not a vowel." Miscellaneous Examples of the Method. We can apply the Indirect Method of Inference however many may be the terms involved or the premises con- taining those terms. As the working of the method is best learnt from examples, I will take a case of two premises forming the syllogism Barbara : thus Iron is metal (i) Metal is element (2) If we want to ascertain what inference is possible concern- ing the term Iron, we develop the term by the Law of Duality. Iron must be either metal or not-metal; iron which is metal must be either element or not-element ; and similarly iron which is not-metal must be either element or not- element. There are then altogether four alternatives among which tlie description of iron must be contained ; thus Iron, metal, element, (a) Iron, metal, not-element, {fi) Iron, not-metal, element, (7) Iron, not-metal, not-element. (3) Our lirst premise informs us that iron is a metal, and if we substitute this description in (7) and (8) we shall have self-contradictory combinations. Our second premise like- wise informs us that metal is element, and applying this description to (yS) we again have self-contradiction, so that there renaains only (a) as a description of iron — our inference is Iron = iron, metal, element. To represent this process of reasoning in general symbok let A = iron B = metal C = element, The premises of the pi-oblem take the forma n.] THE INDIRECT METHOD OF INFERENCE. 80 A = AB (I) B = BO. (2) By the Law of Duality we have A = AB f Aft (5) A = AC .|- Ac. (4) Now, if we insert for A in the second side of (3) its description in (4), we obtain what I shall call the develop- merU of A with respect to B and C, namely A = ABC .| ABc .|. A6C -I- khc. (5) Wlierever the letters A or B appear in the second side of (5) substitute their equivalents given in (i) and (2), and the results stated at full length ai-e A = ABC I- ABCc -|. ABftC I- Al^Cc. The last three alternatives break the Law of Contradiction, so that A = ABC I- o I- o •!• o = ABC. This conclusion is, indeed, no more than we could obtain by the direct process of substitution, that is by substituting for B m (1), its description in (2) as in p. 55 ; it is the characteristic of the Indirect process that it gives all possible logical conclusions, both those which we have previously obtained, and an immense number of others or which the ancient logic took little or no account. From the same premises, for instance, we can obtain a description of the class not-element or c. By the Law of Duality we can develop c into four alternatives, thus c = ABc I- Abe I- aBc I- abc. If we substitute for A and B as before, we get c = ABCc j. ABbc I- aBCc .;• abc, and, stnking out the terms which break the Law of Contradiction, there remains c = abc, or what is not element is also not iron and not metal i ins Indirect Method of Inference thus furnishes a complete solution of the following ^Tohhm— Given any number of logical pi^emises or conditions, required the aescnptum of any class of objects, or of any tmn, as governed by tliose conditions. 1 he steps of the process of inference may thus be concisely stated^ nf '*u^^ ^^^ ^^ ^^ Duality develop the utmost number 01 alternatives which may exist in the description of the D 90 THE PRINCIPLES OF SCIENCE. [OHAP. f? ■/ ! l( * / required class or term as regards the terma involved in the premises. 2. For each term, in these alternatives substitute its description as given in the premises. 3. Strike out every alternative which is then found to break the Law of Contradiction. 4. The remaining terms may be equated to the term in question as the desired description. Mr. VemCs Problem, The need of some logical method more powerful and comprehensive than the old logic of Aristotle is strikingly illustrated by Mr. Venn in his most interesting and able article on Boole's logic* An easy example, originallv got, as he says, by the aid of my method as simply described in the Elementary Lessons in Logic, was proposed in examination and lecture-rooms to some hundred and fifty students as a problem in ordinary logic. It was answered by, at most, five or six of them. It was afterwaixls set, as an example on Boole's method, to a small class who had attended a few lectures on the nature of these symbolic methods. It was readily answered by half Or more of their number. The problem was as follows :— " The members of a board were all of them either bondholders, or shareholders, but not both ; and the bondholders as it happened, were all on the board. What conclusion can be drawn ? " The con- clusion wanted is, "No shareholders are bondholders." Now, as Mr. Venn says, nothing can look simpler than the following reasoning, wJien stated :—'* There can be no bondholders who are shareholders ; for if there were they must be either on the board, or off it. But they are not on it, by the first of the given statements ; nor off it, by the second." Yet from the want of any systematic mode of treating such a question only five or six of some hundred and fifty students could succeed in so simple a problem. »^»nd; a Quarterly Review of Psychology I^n4 PhUoeophy : October, 1876, vol. i. p. 487. ^ *^ ^ ' ▼ij THE INDIRECT METHOD OF INFERENCE. 91 By symbolic statement the problem is instantly solved. Taking A = member of board B = bondholder = shareholder the premises are evidently A = ABc j. A6C B = AB. The class C or shareholders may in respect of A and B be developed into four alternatives, C = ABC .|. AbG I- aBC \ ahG. But substituting for A in the first and for B in the third alternative we get C = ABCc .|. AB5C .|- A^C |. aABG I- obQ. The first, second, and fourth alternatives in the above are self-contradictory combinations, and only the^e; strikin*^ them out there remain ** C = AhQ \ ahG = JC. the required answer. This symbolic reasoning is, I believe, the exact equivalent of Mr. Venn's reasoning, and I do not believe that the result can be attained in a simpler manner. Mr. Venn adds that he could adduce other similar instances, that is, instances showing the necessity of a better logical method. Abbreviation of the Process, Before proceeding to further illustrations of the use of this method, I must point out how much its practical employment can be simplified, and how much more easy It is than would appear from the description. When we want to effect at aU a thorough solution of a locrical problem it is best to form, in the first place, a complete series of all the combinations of terms involved in it If there be two terms A and B, the utmost variety of combinations in which they can appear are AB aB llie term A appears in the first and second ; B in the first and third ; a in the thii-d and fourth ; and b in the second and fourth. Now if we have any premise, say A = B, IJ 92 THE PRINCIPLES OF SCIENCE. [CHAF. TI.J THE INDIRECT METHOD OF INFERENCE. 93 1 ,1 f aifi \h we must ascertain which of these combinations will be rendered self-contradictory by substitution; tlie second and third will have to be struck out, and there will remain only AB ha. Hence we draw the following inferences A = AB, B = AB, a^ab, h ^ ah. Exactly the same method must be followed when a question involves a greater number of terms. Thus by the Law of Duality the three terms A, B, C, give rise to eight conceivable combinations, namely ABC (a) aBC (e) ABc (fi) aWc (f) A6C (7) ahO irj) Ahc (5) ahc. (0) The development of the term A is formed by the first four of these; for B we must select (a), (J3), (c), (f); C consists of (a), (7), (e), (17) ; h of (7), (8), (rj\ {0), and so on. Now if we want to investigate complet<}ly the meaning of the premises A = AB (i) B = BC (2) we examine each of the eight combinations as regards each premise; (7) and (3) are contradicted by (i), and (fi) and (f ) by (2), so that there remain only ABC (a) aBC (c) ahC (ff) ahc. • (0) To describe any term under the conditions of the premises (i) and (2), we have simply to draw out the proper com- binations from this list; thus, A is represented only by ABC, that is to say A = ABC, similarly c = ahc. For B we have two alternatives thus stated, B = ABC i aBC ; and for h we have h = ahC •{' ahc. When we have a problem involving four distinct terms we need to double the number of combinations, and as we add each new term the combinations become twii*^ as numerous. Thus produce four combinations eight sixteen thirty-two sixty-four M n »» »» A, B A, 1^, C, A, B, C, D A, B, C, D, E A, B, C, D, E, F and so on. I pix)pose to cfill any such series of combinations the Logtml Alp/iahet, It holds in logical science a position the importance of which cannot be exaggerated, and as we proceed from logical to mathematical considerations it wiU become apparent that there is a close connection between these combinations and the fundamental theorems ot mathematical science. For the convenience of the reader who may wisli to employ the Alpliahet in logical questions, I liave had printed on the next page a complete senes of the combinations up to those of six terms At the very conimencenient, in the first column, is placed a smgle letter X. which might seem to be superfluous. This letter serves to denote that it is always some higher class which IS divided up. Thus the combination AB really means ABX, or that part of some larger class, say X which has the qualities of A and B present. The letter A IS omitted in the greater part of the table merely for the sake of brevity and clearness. In a later chapter on Com- binations It will become apparent that the introduction of tins unit class is requisite in order to complete the analogy with the Arithmetical Triangle there described. I he reader ought to bear in mind that though the Logical Alphabet seems to give mere lists of combinations, these combinations are intended in every case to constitute the development of a term of a proposition. Thus the four combinations AB, A6, aB, ah really mean that any class X is described by the following proposition , X = XAB .|. XAb .|. XaB .|. Xah. If we select the A's, we obtain the foUowing proposition _,. ^ AX = XAB .|. XA6. ihus whatever group of combinations we treat must be n2Zf ^K.rL'^^ \^^^'' "^'-^^^ ^^^^^ ff^^ or umverse symbo ised in the term X ; but, bearing this in mind It 13 needless to complicate our formulae by always mtroducing the letter. All inference consists in passing »rom propositions to propositions, and combinations^^ ^ J| 4k.^ MMM Ilfiis THE PRINCIPLES OF SCIENCE. [cf a9. have no meaning. They are consequently to be regarded in all cases as forming parts of propositions. The Logical Alphabet. I. X n. m IV. ▼. ▼1. M A B ABC ABCD ABCDB a JL A b A B e A B Cd ABC De «. B A bC A B e D ABCdB • h Abe A B ed A B C d< fc U O Aft C D ABcD B a B e A b C d A Be D < « ft C A b e D A Bed E a b 9 Abed aDC D aB Cd a B e D a B e d o !» «" J> a ft d n ft c i> rt ft - d A U ed « AbCD B A bC D« A b C d B Ab C d f Abe D B A b e D < A be d B A b e d « aBCDB a n C 1) « aBC d B a U Cd « ABe D B II Jl e D < « Be d B A R e d e « ft C D B u ( < • ft Cd B • ft C d < abe D B a 6 r D < akt d ¥, i fit. A BCDEP ABODE/ A B C I) e P A B C D e/ A B C d E K A BC d E/ A BC de P A B C d e / A Be D E K ABe D E/ A Be De P A B c D e/ A Bed E P A Be d E/ A B ede P A B e d e / AbCDB P AbC D E/ AbC De P A bC D «/ AbCd E P AbCdE/ Ab Cd e F AbC dt f AbcD E P A be D E/ A beD « P A b e D « / Abed E P Abed K / A b c d e P A b e d < / a B C D E P aBCDB/ aBO De P a B D e / aBC d BP a B Cd B/ a BCd e P a R C d e / aBe DE P •« Be D B/ a B e D « P « B e D e / a B e d E K IT B e d K / a B e d e P a B e d e / a b C D E F ■ bC D B/ nb C D • P « b C D e / abC d K P A b C d E/ a b C d e P a b C d « / a B e D E P a b e D B / « b e D e P a b e D < / a b e d E P ^h e d E / aft e d « P a ft e d « / "1 J:?12™raV»ETH0DOTJNmEl.C£ „ of such combinations whe?, ,,!flf • I ^ *u " """"^^ ?ilrs TdF^ =^ ats^^ :^ deep lo^cal importance 3 ^^ fTTh^TT^ '''^ the symbol of identity and hanno'nv l,„ J^ -1% "^^ number two as the ori.-in nf 1^ ^' ''^ described the dive«ity. division and^pa^tfo''^^' "' '^'. ^y""''"' »' the reWy,, was also reSd hv . • '^ """''^'' *■«»"•' »•■ elements o/ ;xistence! K iK"ted Zl "' *'^ ''i""^' virtue whence come all combSon, T„ generating golden verses ascribed to TvZir! K ^""^ °^ '''« pupa to be virtuous : ' 'J^hagorsH, he conjures his "tL P? "h" °^"'P' ^*« ■*'<'«'• "IX"' tlie Mind rA« «,„r. tl,e fount of Nature', en.lleas S." Now four and the higher nowpr* nf .i...i-* j in this logical svstem th7n,l^7 i '*'''y ''•' "^Present can be SnS in fhl T^*^ '''l?"''*"*'«"n«^'Wcl. The fol W« S^Pythatoras' mfv K '"S''?^ restrictions, master's doctrines in raSL™ ^-, *''* '^'^''^^ "'«*'• but in many w)i,faT!!T^,'^"'* superstitious notions, basis in logiiEi;'!,^ •^^*""«« ^eem to have some Tkfi Logical State. sigli^i'carcrand '"tiiuvT T^f^^'''^ ^^e ext:.me • indirect proems Tllj! *? ^^^^ Alphabet the repetition'^^a few n^^^^^ "^T"* '^^''^ ^ the selection, and e 'mina£ nf^"* 'T ^'^ "'"s^ification. deduction, even Tn the „1,"°"*'*<^,«''^"««- I^ieal becomes k mat er of ZrTrLT^^T^ ^««««'»»«. mere routine, and the amount of ' Wl..well, mi^ of iU Inductive SoUnce., vol. 1. p. „^ 'J :• • r' ■•- > ': THE PRINCIPLES OF SCIENCE. [cBAf. labour required is the only impediment, when once the meaning of the premises is rendered clear. But the amount of labour is often found to be considerable. The mere writing down of sixty-four combinations of six letters each is no small task, and, if we had a problem of five premises, each of the sixty-four combinations would have to be examined in connection with each premise. The requisite comparison is often of a very tedious character, and considerable chance of error intervenes. I have given much attention, therefore, to lessening both the manual and mental labour of the process, and I shall describe several devices which may be adopted for saving trouble and risk of mistake. In the first place, as the same sets of combinations occur over and over again in different problems, we may avoid the labour of writing them out by having the sets of letters ready printed upon small sheets of \vriting-paper. It has also been suggested by a correspondent that, if any one series of combinations were marked upon the margin of a sheet of paper, and a slit cut between each pair of combinations, it would be easy to fold down any particular combination, and thus strike it out of view. The com- binations consistent with the premises would then remain in a broken series. This method answers sufficiently well for occasional use. A more convenient mode, however, is to have the series of letters shown on p. 94, engraved upon a common school writing slate, of such a size, that the letters may occupy only about a third of the space on the left hand side of the slate. The conditions of the problem can then be written down on the unoccupied part of the slate, and the proper series of combinations being chosen, the contra- dictory combinations can be struck out with the pencil. I have used a slate of this kind, which I call a Logical Slate, for more than twelve years, and it has saved me much trouble. It is hardly possible to apply this process to problems of more than six terms, owing to the large number of combinations which would require examination. fi.) THE INDIKECT METHOD OF INFEKENCE. Vt Abstraction of Indifferent Circumstances, There is a simple but highly important process of inference which enables us to absti-act, eliminate or dis- regard all circumstances indifferently present and absent Thus if I were to state that " a triangle is a three-sided rectilinear figure, either large or not lai^e," these two alternatives would be superfluous, because, by the Law of Duality, I know that everything must be either large or notlai^e. To add the qualification gives no new know- ledge, since the existence of the two alternatives wiU be understood in the absence of any information to the contrary. Accordingly, when two alternatives differ only as regards a single component term which is positive in one and negative in the other, we may reduce them to one term by striking out their indifferent part. It is really a process of substitution which enables us to do tliis • for having any proposition of Ihe form ' A = ABC J. ABc, (I) we know by the Law of Duality that , , AB=ABC .|. ABc. (2) As the second member of this is identical witd the s^ond member of (i) we may substitute, obtaining A = AB. This process of reducing .useless alternatives may bo applied again and again ; for it is plain that A = AB (CD .|. Qd .|- cD .|. cd) cfimmunicates no more information than that A is B Abstraction of indifferent terms is in fact the converse process to that of development described in p. 89; and It IS one of the most important operations in the whole sphere of reasoning. The reader should observe that in the proposition AC = BC we cannot abstract C and infer but from AC ^. Ac = BC 1- Be we may abstract all reference to the term C. It ought to be carefully remarked, however, that alter- natives which seem to be without meaning often imply important knowledge. Thus if I say that " a triangle w a u ; . 98 THE PRINCIPLES OF SCIENCE. fOHAP. Fi.] THE INDIRECT METHOD OP INFERENCE. 99 ^ H I;! i it! three-sided rectilinear figure, with or without three equal angles," the last alternatives really express a property of triangles, namely, that some triangles have three equal angles, and some do not have them. If we put P =» " Some," meaning by the indefinite adjective " Some," one or more of the undefined properties of triangles with three equal angles, and take A = triangle B = three-sided rectilinear figure, C = with three equal angles, then the knowledge implied is expressed in the two propositions PA = PBC joA = pBc, These may also be thrown into the form of one pro- position, namely, A = PBC I- pBc; but these alternatives cannot be reduced, and the propo- sition is quite different from A = BC I- Be Jllustrations of the Indirect Method. A great variety of arguments and logical problems might be introduced here to show the comprehensive character and powers of the Indirect Method. We can treat either a single premise or a series of premises. Take in the first place a simple definition, such as " a triangle is a three-sided rectilinear figure." Let A = triangle B =s three-sided C "= rectilinear figure ; then the definition is of the form A = BC. If we take the series of eight combinations of three letters in the Logical Alphabet (p. 94) and strike out those which are inconsistent with the definition, we have ♦JiP following result : — ABC dBc ahC For the description of the class C we have C = ABC A' ahC, that is, " a rectilinear figure is either a triangle and three- sided, or not a triangle and not three-sided." For the class b we have h = ahC •(• abc. To the second side of this we may apply the profr<»ss of simplification by abstraction described in the last section • for by the Law of Duality * ah = abC •(• abc ; and as we have two propositions identical in the second side of each we may substitute, getting h — aby or what is not three-sided is not a triangle (wliether it be rectilinear or not). Second Example, H! S^ ^^^\^y ^^^ method the following ai^ument :— " Blende is not an elementary substance ; elementary substances are those which are undecomposable ; blende, therefore, \& decomposable." Taking our letters thus— A = blende, « B = elementary substance, C = undecomposable, the premises are of the forms A = AJ, (,) B = C. (2) No immediate substitution can be made ; but if we take lUe contrapositive of (2) (see p. ^6), namely i = c, /^\ we can substitute in (i) obtaining the conclusion mrnhf ^•^°'® result may bl obtained by taking the eight h^t^'^A'?^.''^ .^' ?' ^' ^^ *^^ Logical Alphabet; it wiU DC lound that only three combinations, namely Abe . aBC abc, are consistent with the premises, whence it results thai A = Abc, H 2 ? I r I 100 THE PRINCIPLES OF SCIENCE. [OIIAP. or by the process of Ellipsis before described (p. 57) A = Ac. Third Example, As a somewhat more complex example I take the argument thus stated, one which could not be thrown into the syllogistic form : — " All metals except gold and silver are opaque ; there- fore what is not opaque is either gold or silver or is not-metal." There is more implied in this statement than is dis^ linctly asserted, the full meaning being as follows: (I) (2) (3) (4) (I) (2) (3) (45 All metals not gold or silver are opaque, Gold is not opaque but is a metal, Silver is not opaque but is a metal, Gold is not silver. Taking our letters thus — A = metal C = silver ^ = gold J) = opaque, we may state the premises in the forms Abe = AhcD B = AB</ C = ACd B = Be. To obtain a complete solution of the question we Take the sixteen combinations of A, B, C, D, and striking out those which are inconsistent with the premises, there i-emain only ABcd AhCd AhcD abcD ; abed. The expression for not-opaque things consists of the three combinations containing d, thus d==ABed •]• AbCd .|. abed, or rf " A<£ (Be .|. bC) + al)cd. In ordiuar}' language, what is not-opaque is either metal which is gold, and then not-silver, or silver and then not gold, or else it is not-metal and neither gold nor silver. VI.] THE INDTREOl' METHOD OF INFERENCE. loi Fourth Example. A good example for the illustration of the Indirect Method is to be found in De Morgan's Formal Logic (p 123), the premises being substantially as follows :-~ From A follows B, and from follows D ; but B and D are inconsistent with each other ; therefore A and C are inconsistent. The meaning no doubt is that where A is, B will be found, or that every A is a B, and similarly eveiy C is a D • but B and D cannot occur together. The premises there- fore appear to be of the forms A = AB, (,) C = CD, 2 B = Brf. ).{ On examining the series of sixteen combinations, only five are found to be consistent with the above condition? namely, ABcd aBed abOD abcT> abed. In these combinations the only A which appears is joined with C ^^""'^^'■^^ ^ ^^ J^'"^^ *^ a, or A is inconsistent Fifth Example. A more complex argument, also given by De Morgan i contjains five t^rms, and is as stated below, except that the letters are altered. » ^ * Every A is one only of the two B or C ; D is both B and C, except when B is E, and then it is neither ; therefore no A is D. The meaning of the above premises is difficult to interpret, but seems to be capable of expression in the foUowmg symbolic forms— PointenSi ^' ?l '^^ ^^ Professor Croom Robrt^on has mto T.i«^* ^^ ™^? ^^^ S^^'**^ *°^ ^^"^ premises out be thrown into a single propontion, D - D#BO f DKic iwrown I' I H lot THE PRINCIPLES OF SCIENCE. [OHAT. ABcdE ABcde AhCdE AbCde ahCdE abCde ahcDE ahcdE abode. A = AB(; I- AbQ, (i) De = DeBC, (2) DE=DE^»c. (3) As five terms enter into these premises it is requisite to treat their thirty-two combinations, and it will be found that fourteen of them remain consistent with the premises namely ' aBCD<5 a^Gde aBcrfE dEcde If we examine the first four combinations, all of which contain A, we find that they none of them contain D ; or again, if we select those which contain D, we have only two, thus — D = aBCD(5 .[. alcDE. Hence it is clear that no A is D, and vice versd no D is A. We might draw many other conclusions from the same premises ; for instance — DE = ahcDE, or D and E never meet but in the absence of A, B, and C. Fallacies analysed by the Indirect Method. It has been sufficiently shown, perhaps, that we can by the Indirect Method of Inference extract the whole truth fron) a series of propositions, and exhibit it anew in any required form of conclusion. But it may also need to be shown by examples that so long as we follow conectly the almost mechanical rules of the method, we cannot fall into any of the fallacies or paralogisms which are often committed in ordinary discussion. Let us take the example of a fallacious argument, previously treated by the Method of Direct Inference (p. 62), Granite is not a sedimentaiy rock, (i Basalt is not a sedimentary rock, (21 and let us ascertain whether any precise conclusion can be drawn concerning the relation of granite and basalt Taking as before A = granite, B = sedimentary rock, C =» basalt. lu] THE INDIRECT METHOD OF INFERENCE. 103 the premises become A^ Ah (i) C = C^' (2) Of the eight conceivable combinations of A, B. 0. five agree with these conditions, namely AbG aBe Abe ahC abc. Selecting the combinations which contain A, we find the description of granite to be A = A5C |. Abc=^Ab(G •(. c), that IS, granite is not a sedimentary rock, and is either basalt or not-basalt. If we want a description of basalt the answer is of like form C = A^C .|. abG = bC(A + a), that 18 basalt is not a sedimentary rock, and is either granite or not-granite. As it is ahready perfectly evident that basalt must be either granite or not, and vice versd the premises fail to give us any information on the point' that IS to say the Method of Indirect Inference saves us from falling into any fallacious conclusions. This example sufficiently iUustrates both the fallacy of Negative premises and that of Undistributed Middle of the old logic The faUacy called the Illicit Process of the Major Term IS also incapable of commission in following the rules of the method. Our example was (p. 65) All planets are subject to gravity, (i) Fixed stars are not planets. (2) The false conclusion is that " fixed stars are not subject to gravity." The terms are ^ A = planet B = fixed star - , ^_ C = subject to gravity. And the premises are A = AC, (i) -^ B = aB. (2) The combinations which remain uncontradicted on com- parison with these premises are AbG aBe aBG abG For fixed star we have the description B = aBC A- aBc t fmi 104 THE PRINC IPLES OP SCIENCE. [chap. f ni^Ll ,. *i, **" *' ?"' * P''^"^'- ^"' i« «*« subject or not. as the case may be, to gravity." Here we have no conclusion concerning the connection of fixed stars and gravity. " The Logical Abaeut. _ The Indirect Method of Inference lias now been suffi ciently described, and a careful examina^on of ^ — Sfchief diSu'^''^''?? '"^°^r"." °"'y >««*<='*» ^J^tions ine chief difficulty of the method consists in the great number of combinations which may have to be exami^ not on^y may the requisite labour become form'dabTe but a considerable chance of mistake arises. I hive the^foS given much attention to modes of facilitating the work miharafT™ '''t^ ^"""^ the method t^ an alS mechanical form It soon appeared obvious that if the conceivable combinations of the Logical Alphabet for anv T^l^'fr- ^''^^ f being^printed^" fix'ed onTr ^n^oKp^ PfP*' °' ='«•*• '^ere marked upon lieht ITi iP'r '? "/ 7'^' niechanical armngemeSts could St thr ;<. ^* ^*^"' "I comparison and rejection Si in fh- if ^^'^ "^i"^"*"' ^'''<='' I have found rXln nf w- Y"1"r""" ^""^ exhibiting the complete solution of logical problems. A minute description of the oo.«truction and use of the Abacus, together wiXfigures mlZT^^ ^'^^^.,heen given if my essay cE JAe bubdUutum of Simtiars.' and I will here cive onlv a general description. *» ' ho^l^S*' ^hacus «)nsists of a common school black- board placed in a sloping position and furnished with four of thTSte^^shT"^'^^ ^^«^- ^""^ cornhZtZ lUnfi if^ t?""" '" *he first four columns of the Logical Alphabet are printed in somewhat lame type 80 that each letter is about an inch from the neTh W inst^ of being m horizontal lines as in p at. Each combumtion of letters is sepamtely fixed to the^urface of ' I> SS-S* 81—86. 11 VI.] THE INDIRECT METHOD OF INFERENCK 105 a thin slip of wood one inch broad and about one-eighth inch thick Short steel pins are then driven in an inclined position into the wood. When a letter is a large capital representing a positive term, the pin is fixed in the upper part of its space ; when the letter is a small italic repre- senting a negative term, the pin is fixed in the lower part of the space. Now, if one of the series of combinations be ranged upon a ledge of the black-board, the sharp edge of a flat rule can be inserted beneath the pins belonging to xny one letter — say A, so that all the combinations marked A can be lifted out and placed upon a separate ledge. Thus we have represented the act of thought which separates the class A from what is not- A. The operation can be repeated ; out of the A's we can in like manner select those which are B s, obtaining the AB's ; and in like manner we may select any other classes such as the aB's, the ab*B, or the ahc*8. If now we take the series of eight combinations of the letters A, B, C, a, b, c, and wish to analyse the argument anciently called Barbara, having the premises A = AB (I) B = BC, (2) we proceed as follows — We raise the combinations marked a, leaving the A's behind; out of these A's we move to a lower ledge such as are 6's, and to the remaining AB's we join the a's which have been raised. The result is that we have divided all the combinations into two classes, namely, the Aft's which are incapable of existing consist- ently with premise (i), and the combinations which are consistent with the premise. Turning now to the second premise, we raise out of those which agree with (i) the i's, then we lower the IVs ; lastly we join the 5*s to the BC's. We now find our combinations arranged as below. A a a a B B h b C C c e A A A a B h b B e C e e The lower line contains all the combinations which are inconsistent with either premise ; we have carried out in a ,1^ ; 1 106 THE PRINCIPLES OF SCIENCE. [iMAT, mechanical manner that exclusion of self-contradictories which was formerly done upon the slate or upon paper Accordingly, from the combinations remaining in the upper line we can draw any inference which the premises yield If we raise the A's we find only one, and that is C so that A must be C. If we select the c s we again find only one, which is a and also 3 ; thus we prove that not-C is not-A and not-B. When a disjunctive proposition occurs among the premises the requisite movements become rather more complicated. Take the disjunctive argument A is either B or C or D, A is not C and not D, Therefore A is B. The premises are represented accurately as follows :— A = AB t AC I- AD (I) A = Ad. /^\ As there are four terms, we choose the series of sixteen combinations and place them on the highest ledffe of the board but one. We raise the a's and out of the A's. which remain, we lower the b's. But we are not to reject all the A6 s as contradictory, because by the first premise A's may be either Bs or C's or D's. Accordingly out of the Ah 8 we must select the c% and out of these again the rf's 80 that only Abed will remain to be rejected finally! Joining all the other fifteen combinations together aaain and proceeding to premise (2), we raise the a's and iSwer the AGs, and thus reject the combinations inconsistent with (2) ; similarly we reject the AD's which are incon- sistent with (3) It will be found that there remain, in addition to all the eight combinations containing a only one contiiining A, namely ' ABcd, whence it is apparent that A must be B, the ordinary conclusion of the argument. In my "Substitution of Similars" (pp. 56—50) I have described the working upon the Abacus of two other logical problems, which it would be tedious to repeat in tins place. '^ VI.] THE INDIRECT METHOD OF INFERENCE. 107 The Logical Machine. Although the Logical Abacus considerably reduced the lalwur of using the Indirect Method, it was not free from the possibility of error. I thought moreover that it would affortl a conspicuous proof of the generality and power of the method if I could reduce it to a purely mechanical form. Logicians had long been accustomed to speak of Logic as an Organon or Instrument, and even Lord Bacon, while he rejected the old syllogistic logic, had insisted, in the second aphorism of his " New Instrument," that the mind required some kind of systematic aid. In the kindred science of mathematics mechanical assistance of one kind or another had long been employed. Orreries, globes, mechanical clocks, and such like instruments, are really aids to calculation and are of considerable antiquity. The Arithmetical Abacus is still in common use in Russia and China. The calculating machine of Pascal is more than two centuries old, having been con- structed in 1642-45. M. Thomas of Colmar manufactures an arithmetical machine on Pascal's principles which is employed by engineers and others who need frequently to multiply or divide. To Babbage and Scheutz is due the merit of embodying the Calculus of Differences in a machine, which thus became capable of calculating the most complicated tables of figures. It seemed strange that in the more intricate science of quantity mechanism should be applicable, whereas in the simple science of qualitative reasoning, the syllogism was only called an instrument by a figure of speech. It is true that Swift satirically described the Professors of Laputa as in pos- session of a thinking machine, and in 185 1 Mr. Alfred Smee actually proposed the construction of a Relational machine and a Differential machine, the first of which would be a mechanical dictionary and the second a mode of comparing ideas; but with these exceptions I have not yet met with so much as a suggestion of a reasoning machine. It may be added that Mr. Smee's designs, though highly ingenious, appear to be impracticable, and in . any case they do not attempt the performance of logical inference.^ * See hiB work called The Process of Thought adapted to Words and LanguagSy together with a Description of the Belationcd and Differ- 108 THE PRINCIPLES OP SCIENCE. [chap vl] THE INDIEBCT METHOD OF INFERENCE. 100 The Logical Abacus soon suggested the notion of a Logical Machine, which, after two unsuccessful attempts I succeeded m constructing in a comparatively simple and effective form. The detaUs of the Logical Machine have been fuUy described by the aid of plates in the Philo- sophical Transactions,! and it would be needless to repeat the account of the somewhat intricate movements of the machine in this place. The general appearance of the machine is shown in a plate facing the title-page of this volume. It somewhat resembles a very small upright piano or organ, and has a keyboard containing- twenty-one keys. These keys are of two kinds, sixteen of them representing the terms or letters A a, B, 5, C, c, D, d, which have so often been employed in our logical notation. When letters occur on the left-hand side of a proposition, formerly called the subject each is represented by a key on the left-hand half of the keyboard ; but when they occur on the right-hand side, or as it used to be called the predicate of the pro- position the letter-keys on the right-hand side of the keyboard are the proper representatives. The five other keys may be called operation keys, to distinguish them trom the letter or term keys. They stand for the stops copula,, and disjunctive conjunctions of a proposition. The middle key of all is the copida, to be pressed when the verb is or the sign = is met. The key to the extreme nght-hand is caUed the Full Stop, because it should be pressed when a proposition is completed, in fact in the proper place of the full stop. The key to the extreme lelt-hand is used to terminate an ai^ument or to restore the machine to its initial condition ; it is called the Finis key. The last keys but one on the right and left com- plete the whole series, and represent the conjunction or in Its unexclusive meaning, or the sign H which I have employed, according as it occurs in the right or left hand side of the proposition. The whole keyboard is arranged as shown on the next page — mtialMaehinei, Alao PhUotophical Trantactioin, [1870] vol. 160. of Q^ B^al Socuijf, vol. xvui. p. 166, Jan. 20 djo. iyTolm, vol.1 & LefUhind ikl« of Proposition. 1 Right-haud side of Proposition. 1 99 i. 4 D « C b B a A A a B h C e D d Or To work the machine it is only requisite to press the keys in succession as indicated by the letters and signs of a symbolical proposition. All the premises of an argu- ment are supposed to be reduced to the simple notation which has been employed in the previous pages. Taking then such a simple proposition as A = AB, we press the keys A (left), copula, A (right), B (right), and full stop. If there be a second premise, for instance B = BC, we press in like manner the keys — B (left), copula, B (right), C (right), full stop. The process is exactly the same however numerous the premises may be. When they are completed the operator will see indicated on the face of the machine the exact combinations of letters which are consistent with the premises according to the principles of thought. ' As shown in the figure opposite the title-page, the machine exhibits in front a Logical Alphabet of sixteen combinations, exactly like that of the Abacus, except that the letters of each combination are separated by a certain interval. After the above problem has been worked upon the machine the Logical Alphabet will have been modified so as to present the following appearance — *IH 1 .l. a\ a a a B B 1 1 b|b| \b h b b 1 |o|c C C « 1 c' • 1 0|4 1 1 D d D d DJ d )' no THE PRINCIPLES OF SCIENOE. [chap. Ti.] THE INDIRECT METHOD OF INFERENCE. Ill I ^ The operator will readily collect the various conclusions m the manner described in previous pages, as for in- stance that A is always C, that not-C is not-B and not- A ; and not-B is not-A but either C or not-C. The results are thus to be read off exactly as in the case of the ijogical Slate, or the Logical Abacus. Disjunctive propositions are to be treated in an exactly similar manner. Thus, to work the premises A = AB .|. AC i B + C = BD.|.CD, It IS only necessary to press in succession the keys T> A.^J^^9' ^P""^^' A ^"Sht), B, .|. , A,C, full stop. B aeft). .|. C, copula, B (right). D, + , C,D, full stop. Ihe combinations then remaining will be as follows ABCD aBCt) abd) ABcB aBcB abed. AbCD abCB On pressing the left-hand key A, aU the possible com- bmations which do not contain A will disappear, and the description of A may be gathered from what remain namely that it is always D. The full-stop key restores aU combinations consistent with the premises and any other selection may be made, as say not-D, which wUl be found to be always not-A, not-B, and not-C. At the end of every problem, when no further questions need be addressed to the machine, we press the Finis key, which has the effect of bringing into view the whole ot the conceivable combinations of the alphabet This key in fact obliterates the conditions impressed upon the machine by moving back into their ordinary places those combinations which had been rejected as inconsistent with the premises. Before beginning any new problem it is requisite to observe that the whole sixteen combinations are visible. After the Finis key has been used the machine represents a mmd endowed with powers of thought, but whoUy devoid of knowledge. It would not in that con- dition give any answer but such as would consist in the primary laws of thought themselves. But when any pro- position IS worked upon the keys, the machine analyses aiid digests the meaning of it and becomes charged with the knowledge embodied in that proposition. Accordingly It 18 able to return as an answer any description of a term or class so far as furnished by that proposition in accordance with the Laws of Thought. The machine is thus the em- bodiment of a true logical system. The combinations are classified, selected or rejected, just as they should be by a reasoning mind, so that at each step in a problem, the Logical Alphabet represents the proper condition of a mind exempt from mistake. It cannot be asserted indeed that the machine entirely supersedes the agency of conscious thought; mental labour is required in interpreting the meaning of grammatical expressions, and in con*ectly im- pressing that meaning on the machine ; it is further required in gathering the conclusion from the remaining combina- tions. Nevertheless the true process of logicsd inference is really accomplished in a purely mechanical manner. It is worthy of remark that the machine can detect any self-contradiction existing between the premises presented to it ; should the premises be self-contradictory it will be found that one or more of the letter-terms disappears entirely from the Logical Alphabet. Thus if we work the two propositions, A is B. and A is not-B, and then inquire for a description of A, the machine will refuse to give it by exhibiting no combination at all containing A. This result is in agreement with the law, which I have ex- plained, that every term must have its negative (p. 74). Accordingly, whenever any one of the letters A, B, C, D, a, 6, c, d, wholly disappears from the alphabet, it may be safely inferred that some act of self-contradiction has been committed. It ought to be carefully observed that the logical machine cannot receive a simple identity of the form A — B except in the double form of A = B and B = A. To work the proposition A = B, it is therefore necessary to press the keys — A (left), copula, B (rights full stop ; B (left), copula, A (right), full stop. The same double operation will be necessary whenever the proposition is not of the kind called a partial identity (p. 40). Thus AB = CD, AB = AC, A = B i C, A j- B = C .|. D, all require to be read from both ends separately. The proper rule for using the machine may in fact be given in the following way : — (i) Bead each proposition as it UandSf and play ihe corresponding keys : (2) Convert tJu 1^1' I 112 THE PRINCIPLES OF SCIENCE. [CDAP. fu] THE INDIRECT METHOD OF INFERENCE. 113 .Ji' proposition and read atid play the keys again in the trans- posed order of the terms. So long as this rule is observed the true result must always be obtained. There can be no mistake. But it will be found that in the case of partial identities, and some other similar forms of propositions, the transposed reading has no effect upon the combinations of the Ix^ical Alphabet. One reading is in sucli cases all that is practically needful After some experience has been gained in the use of the machine, the worker naturally saves himself the trouble of the second reading when possible. It is no doubt a remarkable fact that a simple identity cannot be impressed upon the machine except in the form of two partial identities, and this may be thought by some logicians to militate against the equational mode of repre- senting propositions. Before leaving the subject I may remark that these mechanical devices are not likely to possess much practical utility. We do not require in common life to be constantly solving complex logical questions. Even in mathematical calculation the ordinary rules of arithmetic are generally sufficient, and a calculating machine can only be used with advantage in peculiar cases. But the machine and abacus have nevertheless two important uses. In the first place I hope that the time is not very far distant when the predominance of the ancient Aristotelian Logic will be a matter of history only, and when the teaching of logic will be placed on a footing more worthy of its supreme importanca It will then be found that the solution of logical questions is an exercise of mind at least as valuable and necessary as mathematical calculation. I believe that these mechanical devices, or something of the same kind, will then become useful for exhibiting to a class of students a clear and visible analysis of logical problems of any degree of complexity, the nature of each step being rendered plain to the eyes of the students. I often used the machine or abacus for this purpose in my class lectures whUe I was Professor of Logic at Owens College. Secondly, the more immediate importance of the machine seems to consist in the unquestionable proof which it affords that correct views of the fundamental principles of reasoning have now been attained, although they were unknown to Aristotle and his followers. The time must come when the inevitable results of the admirable investigations of the late Dr. Boole must be recognised at their true value, and the plain and palpable form in which the machine piesents those results will, I hope, hasten the time. Undoubtedly Boole's life marks an era in the science of human reason. It may seem stranj^e that it had remained for him first to set forth in its full extent the problem of logic, but I am not aware that anyone before him had treated logic as a symbolic method for evolving from any premises the description of any class whatsoever as defined by those premises. In spite of several serious errors into which he fell, it will probably be allowed that Boole discovered the true and general form of logic, and put the science substantially into the form which it must hold for evermore. He thus effected a reform with which there is hardly anything comparable in the history of logic between his time and the remote age of Aristotle. Nevertheless, Boole's quasi* mathematical system could hardly be regarded as a final and unexceptionable solution of the problem. Not only did it require the manipulation of mathematical symbols in a very intricate and perplexing manner, but the results when obtained were devoid of demonstrative force, because they turned upon the employ- ment of unintelligible symbols, acquiring meaning only by analogy. I have also pointed out that he imported into his system a condition concerning the exclusive nature of alternatives (p. 70), which is not necessarily true of logical terms. I shall have to show in the next chapter that logic is really the basis of the whole science of mathematical reasoning, so that Boole inverted the true order of proof when he proposed to infer logical truths by algebraic processes. It is wonderful evidence of his mental power that by methods fundamentally false he should have succeeded in reaching true conclusions and widening the sphere of reason. The mechanical performance of logical inference affords a demonstration both of the truth of Boole's results and of the mistaken nature of his mode of deducing them. Conclusions which he coiild obtain only by pages of intri- cate calculation, ai-e exhibited by the machine after one or I 114 THE PRINCIPLBS OF SCIENCR. [OUAP. VI.1 THE INDIRECT METHOD OF INFERENCE. 115 two minutes of manipulation. And not only are those conclusions easily reached, but they are demonstratively true, because every step of the process involves nothing more obscure than the three fundamental Laws of Thought The Order of Premises, Before quitting the subject of deductive reasomng, I may remark that the order in which the premises of an argument are placed is a matter of logical indifference. Much discussion has taken place at various times con- cerning the arrangement of the premises of a syllogism ; and it has been generally held, in accordance with the opinion of Aristotle, that the so-called major premise, containing the major term, or the predicate of the con- clusion, should stand first. This distinction however falls to the ground in our system, since the proposition is reduced to an identical form, in which there is no distinc- tion of subject and predicate. In a strictly logical pomt of view the order of statement is wholly devoid of significance. The premises are simultaneously coexistent, and are not related to each other according to the properties of space and time. Just as the qualities of the same object are neither before nor after each other in nature (p. 33), and are only thought of in some one order owing to the' limited capacity of mind, so the premises of an argument are neither before nor after each other, and are only thought of in succession because the mind cannot grasp many ideas at once. The combinations of the logical alphabet are exactly the same in whatever order the premises be treated on the logical slate or machine. Some difference may doubtless exist as regards convenience to human memory. -The mind may take in the results of an argument more easily in one mode of statement than another, although there is no real difference in the logical results. But in this point of view I think that Aristotle and the old logicians were clearly wrong. It is more easy to gather the conclusion that " all A's are C's ' from " all A's are B's and all B's are C's," than from the same propositions in inverted order, " all B's are C's and all A's ai-e B's. The Equivalence of Propositions One great advantage which arises from the study of this Indirect Method of Inference consists in the clear notion which we gain of the Equivalence of Propositions. The older logicians showed how from certain simple premises we might draw an inference, but they failed to point out whether that inference contained the whole, or only a part, of the information embodied in the premises. Any one proposition or group of propositions may be classed with respect to another proposition or qroup of propositions, as 1. Equivalent, 2. InfeiTible, 3. Consistent, 4. Contradictory. Taking the proposition " All men are mortals " as tlic original, then "All immortals are not men" is its equiva- lent ; " Some mortals are men " is infenible, or capable of inference, but is not equivalent ; ** All uot-men are not mortals" cannot be inferred, but is consistent, that is, may be true at the same time ; " All men are immortals " is of course contradictory. One sufficient test of equivalence is capability of mutual inference. Thus from All electrics = all non-conductors, I can infer All non-electrics = all conductors, and vice versd from the latter I can pass back to the former. In short, A = B is equivalent to a = b. Again, from the union of the two propositions, A = AB and B = AB, I get A = B, and from this I might as easily deduce the two with which I started. In this case one proposition is equivalent to two other propositions. There are in fact no less than four modes in which we may express the identity of two classes A and B, namely, HRST MODB. SECOND MODE. THIRD MODE. FOUKTH jIODEl A-B a = J B = Ab} The Indirect Method 01 Inference furnishes a universal and clear criterion as to the i-elationship of propositions. The import of a statement is always to be measured by I 2 JUKTH JtUU a = ab\ b = abf 116 THE PRINCIPLES OF SCIENCE. [oBap the combinations of terras which it destroys. Hence two propositions are equivalent when they remove the same combinations from the Logical Alphabet, and neither more nor less. A proposition is inferrible but not equivalent to another when it removes some but not all the combinations which the other removes, and none except what this other removes. Again, propositions are consistent provided that they jointly allow each term and the negative of each term to remain somewhere in the Logical Alphabet. If after all the combinations inconsistent with two propo- sitions are struck out, there still appears each of the letters A, a, B, h, C, c, D, rf, which were there before, then no inconsistency between the propositions exists, although they may not be equivalent or even inferrible. Finally, contradictory propositions are those which taken together remove any one or more letter-terms from the Logical Alphabet. What is true of single propositions applies also to groups of propositions, however large or complicated ; that is to say, one group may be equivalent, inferrible, consistent, or contradictory as regards another, and we may similarly compare one proposition with a group of propositions. To give in this place illustrations of all the four kinds of relation would require much space : as the examples given in previous sections or chapters may serve more or less to explain the relations of inference, consistency, and contradiction, I will only add a few instances of equivalent propositions or groups. In the following list each proposition or group of pro- j)Ositions is exactly equivalent in meaning to the corre- sponding one in the other column, and the truth of this statement may be tested by working out the combinations of the alphabet, which ought to be found exactly the same in the case of each pair of equivalents. A — b ... A = BC. . . A = AB.|.AC. A + B = C.|D. . A + c = B -I- (i . . A =» ABc I- A5C ! B = aB a = B a = h'\'C h^ab-l' AM) ah ^ cd aC = 5D A = AB I- AC AB«^ ABc fi.] THE INDIKBOT METHOD OF INFBUBNCE. 117 A = B) / A = B B = Cj • • • \A = Q A = AB ) I A = AC B = BC j • • • t B = A I- aBC Although in these and many other cases the equivalents of certain propositions can readily be given, yet I believe that no uniform and infallible process can be pointed out by which the exact equivalents of premises can be ascertained. Ordinary deductive inference usually gives us only a portion of the contained information. It is true that the combinations consistent with a set of premises may always be thrown into the form of a proposition which must be logically equivalent to those premises ; but the difficulty consists in detecting the other forms of propositions which will be equivalent to the premises. The task is here of a different character from any which we have yet attempted. It is in reality an inverse process, and is just as much more troublesome and uncertain than the direct process, as seeking is compared with hiding. Not only may several different answers equally apply, but there is no method of discovering any of those answers except by repeated trial. The problem which we have here met is really that of induction, the inverse of deduction ; and, as I shall soon show, induction is always tentative, and, unless conducted with peculiar skill and insight, must be exceedingly laborious in cases of complexity. De Morgan was unfoitunately led by this equivalence of propositions into the most serious error of his ingenious system of Logic. He held that because the proposition " All A's are all B's," is but another expression for the two propositions " All A's are B*s " and " All B's are A's, it must be a composite and not really an elementary form of proposition.* But on taking a genei*al view of the equivalence of propositions such an objection seems to have no weight. Logicians have, with few exceptions, persistently upheld the original error of Aristotle in rejecting from their science the one simple relation of identity on which all more complex logical relations must really rest • Syllahus of a proposed syftem of Logic, §§ 57, 121, &c F(jfnnm Logic, p. 66^ 118 THE PRINCIPLES OF SCIBNCB. [chap. tri.] THE INDIRECT METHOD OF INFERENCE. lid Lii TJie Nature of Inference, ^ The question, What is Infereoce ? is involved, even to the present day, in as much uncertainty as that ancient question, What is Truth ? I shall in more than one part of this work endeavour to show that inference never does more than explicate, unfold, or develop the information contained in certain premises or facts. Neither in deduc- tive nor inductive i-easoning can we add a tittle to our implicit knowledge, which is like that contained in an unread book or a sealed letter. Sir W. Hamilton has well said, 'I Reasoning is the showing out explicitly that a proposition not granted or supposed, is implicitly contained in something dififerent, which is granted or supposed/' ^ ^Professor Bowen has explained « with much clearness that the conclusion of an argument states explicitly what is virtually or implicitly thought. " The process of reasoning IS not so much a mode of evolving a new truth, as it is of establishing or proving an old one, by showing how much was admitted in the concession of the two premises taken together." It is true that the whole meaning of these statements rests upon that of such words as " explicit " * implicit," " virtual." That is implicit which is wrapped up, and we render it explicit when we unfold it. Just as the conception of a circle involves a hundred important geometrical properties, all following from what we know, if we have acuteness to unfold the results, so every fact and statement involves more meaning than seems at first sight. Reasoning explicates or brings to conscious posses- sion what was before unconscious. It does not create, nor does it destroy, but it transmutes and throws the same matter into a new form. The difficult question still remains, Wliere does novelty of form begin ? Is it a case of inference when we pass from " Sincerity is the parent of truth " to " The parent of truth IS smcerity ?" The old logicians would have called this change conversion, one case of immediate inference. But as all identity is necessarily reciprocal, and the very meanmg of such a proposition is that the two terms aro * Lectures on Metaphysics, vol. iv. p. 369. » Bowen, TreaHse on Logic, Ciuubridge, U.S., 1866 ; p. 362. identical in their signification, I fail to see any differencje between the statements whatever. As well might we say that X = 7/ and y = x are different equations. Another point of difficulty is to decide when a change is merely grammatical and when it involves a real logical transformation. Between a table of wood and a wooden table there is no logical difference (p. 31), the adjective being merely a convenient substitute for the prepositional phrase. But it is uncertain to my mind whether the change from " All men are mortal " to " No men are not nioital" is purely grammatical. Logical change may perhaps be best described as consisting in the determination of a relation between certain classes of objects from a relation between certain other classes. Thus I consider it a truly logical inference when we pass from " All men are mortal" to "AH immortals are not-men," because the clas.sos immortals and not-men are different from mortals and men, and yet the propositions contain at the bottom the very same truth, as shown in the combinations of the Logical Alphabet. The passage from the qualitacive to the quantitative mode of expressing a proposition is another kind of change which we must discriminate from true logical inference. We state the same truth when we say that "mortality belongs to all men," as when we assert that " all men arc mortals." Here we do not pass from class to class, but from one kind of term, the abstract, to another kind, the concrete. But inference probably enters when we pass from either of the above propositions to the assertion that the class of immortal men is zero, or contains no objects. It is of course a question of words to what processes we shall or shall not apply the name " inference," and I have no wish to continue the trifling discussions which have already taken place upon the subject. What we need to do is to define accurately the sense in which we use the word "inference," and to distinguish the relation of in- ferrible propositions from other possible relations. It seems to be sufficient to recognise four modes in which two apparently different propositions may be related. Thus two propositions may be — I. Tautologons or identical, involving the same relation between the same terms and classes, and only differing in 120 THE PRINCIPLBS OF SCIENCE. [chap. vj. khe order of statement ; thus " Victoria is the Queen of England " is tautologuus with " The Queen of England is Victoria." ° 2. Grammatically/ related, when the classes or objects are the same and similarly related, and the only diflference 13 in the words ; thus " Victoria is the Queen of Enoland " 18 grammatically equivalent to "Victoria is England's Queen." ° 3. Equivalents in qualitative and quantitative form the classes being the same, but viewed in a diflferent manner. 4- Logically inferrible, one from the other, or it may be equivalent, when the classes and relations are dififerent but involve the same knowledge of the possible combinations CHAPTER VIL INDUCTION. ^^ We enter in this chapter upon the second great de- partment of logical method, that of Induction or the Inference of general from particular truths. It cannot be said that the Inductive process is of greater importance than the Deductive process already considered, because the latter process is absolutely essential to the existence of the former. Each is the complement and counterpart of the other. The principles of thought and existence which underlie them are at the bottom the same, just as subtrac- tion of numbei-s necessarily rests upon the same principles as addition. Induction is, in fact, the inverse operation of deduction, and cannot be conceived to exist without the corresponding operation, so that the question of re- lative importance cannot arise. Who thinks of asking whether addition or subtraction is the more important process in arithmetic? But at the same time much difference in difficulty may exist between a direct and inverse operation; the integral calculus, for instance, is infinitely more difficult than the differential calculus of which it is the inverse. Similarly, it must be allowed that inductive investigations are of a far higher degree of difficulty and complexity than any questions of deduction ; and it is this fact no doubt which led some logicians, such as Francis Bacon, Locke, and J. S. Mill, to erroneous^ opinions concerning the exclusive importance of induction. Hitherto we have been engaged in considering how from certain conditions, laws, or identities governing the com- binations of qualities, we may deduce the nature of the 122 THE PRINCIPLES OF SCIENCE. [en A p. combinations agreeing with those conditions. Our work has been to unfold the results of what is contained in any statements, and the process has been one of Synthesis. The terms or combinations of which the character has been determined have usually, though by no means always, involved more qualities, and therefore, by the relation of extension and intension, fewer objects than the terms in which they were described. The truths inferred were thus usually less general than the truths from wliich they were inferred. In induction all is inverted. The truths to be ascer- tained are more general than the data from which they are drawn. The process by which they are reached is analytical^ and consists in separating the complex com- binations in which natural phenomena are presented to us, and determining the relations of separate qualities. Given events obeying certain unknown laws, we have to discover the laws obeyed. Instead of the comparatively easy task of finding what effects will follow from a given law, the effects are now given and tlie law is required. We have to interpret the will by which the conditions of creation were laid down. Induction an Inverse Operation I have already asserted that induction is the inverse operation of deduction, but the difference is one of such great importance that I must dwell upon it. There are many cases in which we can easily and infallibly do a certain thing but may have much trouble in undoing it. A person may walk into the most complicated labyrinth or the most extensive catacombs, and turn hither and thither at his will ; it is when he wishes to return that doubt and difficulty commence. In entering, any path served him ; in leaving, he must select certain definite paths, and in this selection he must either trust to memory of the way he entered or else make an exhaustive trial of all possible ways. The explorer entering a new country makes sure his line of return by barking the trees. The same difficulty arises in many scientific processes. Given any two numbers, we may by a simple and infallible process obtain their product ; but when a large number VII.] INDUCTION. 123 is given it is quite another matter to determine its factors. Can the reader say what two numbers multiplied together will produce the number 8,616460,799? I think it unlikely that anyone but myself will ever know; for they are two large prime numbers, and can only be re- discovered by trying in succession a long series of prime divisors until the right one be fallen upon. The work would probably occupy a good computer for many weeks, but it did not occupy me many minutes to multiply the two factors together. Similarly there is no direct process for discovering whether any number is a prime or not ; it is only by exhaustively trying all inferior numbers which could be divisors, that we can show there is none, and the labour of the process would be intolerable were it not per- formed systematically once for all in the process known as the Sieve of Eratosthenes, the results being registered in tables of prime numbers. The immense difficulties which are encountered in the solution of algebraic equations afford another illustration. Given any algebraic factors, we can easily and infallibly arrive at the product ; but given a product it is a matter of infinite difficulty to resolve it into factors. Given any series of quantities however numerous, there is very little trouble in making an equation which shall have those quantities as roots. Let a, 6, c, d, &c., be the quantities ; then (x — a) (x — h) (x — c) (x - d) = o is the equation required, and we only need to multiply out the expression on the left hand by ordinary mles. But having given a complex algebraic expression equated to zero, it is a matter of exceeding difficulty to discover all the roots. Mathematicians have exhausted their highest powers in carrying the complete solution up to the fourth degree. In every other mathematical operation the inverse process is far more difficult than the direct process, sub- traction than addition, division than multiplication, evo- lution than involution ; but the difficulty increases vastly as the process becomes more complex. Differentiation, the direct process, is always capable of performance by fixed rules, but as tliese rules produce considerable variety of results, the inverse process of integration presents im- mense difficulties, and in an infinite majority of cases surpasses the oresent resources of mathematicians. There MP«'I< I i I * 124 THE PRINCIPLES OF SCIENCE. [chap. are no infallible and general rules for its accomplishment • It must be done by trial, by guesswork, or by remembering the results of differentiation, and using them as a guide Coming more nearly to our own immediate subject exactly the same difficulty exists in determining the law which certain things obey. Given a general mathematical expression, we can infallibly ascertain its value for anv required value of the variable. But I am not aware that mathematicians have ever attempted to lay down the rules of a process by which, having given certain numbers, one might discover a rational or precise formula from which they proceed. The reader may test his power of detectin^r a law, by contemplation of its results, if he, not bein^ a mathematicmn, will attempt to point out the law obeyed by the following numbers : 30' 4a' 30' 5 66' 695^ 2730' 7 6' 3617 510 ' 43867 1^' ete. These numbers are sometimes in low terms but un expectedly spring up to high terms; in absolute magnitude they are very variable. They seem to set all regularity and method at defiance, and it is hardly to be supposed that anyone could, from contemplation of the numbers have detected the relations between them. Yet they are derived from the most regular and symmetrical laws of relation, and are of the highest importance in mathematical analysis, being known as the numbers of Bernoulli Compare again the difficulty of decyphering with that ot cypliering. Anyone can invent a secret language, and with a little steady labour can translate the longest letter into the character. But to decypher the letter, having no key to the signs adopted, is a wholly different matter. As the possible modes of secret writing are infinite in number and exceedingly various in kind, there is no direct mode of discovery whatever. Repeated trial, guided more or less by knowledge of the customary form of cypher and resting entirely on the principles of probability and logical induction, is the only resource. A i)eculiar tact or skUl is requisite for the process, and a few men, such as Wallis or Wheatstone, have attained great success. Induction is the decyphering of the hidden meanincr of natural phenomena. Given events which happen in certain i^i fir.] INDUCTION. 126 definite combinations, we are required to point out the laws which govern those combinations. Any laws being supposed, we can, with ease and certainty, decide whether the phenomena obey those laws. But the laws which may exist are infinite in variety, so that the chances are im- mensely against mere random guessing. The difficulty is much increased by the fact that several laws will usually be in operation at the same time, the effects of which are complicated together. The only modes of discovery consist either in exhaustively trying a great number of supposed laws, a process which is exhaustive in more senses than one, or else in carefully contemplating the effects, endeavouring to remember cases in which like effects followed from known laws. In whatever manner we accomplish the discovery, it must be done by the more or less conscious application of the direct process of deduction. The Logical Alphabet illustrates induction as well as deduction. In considering the Indirect Process of Inference we found that from certain propositions we could infallibly determine the combinations of terms agreeing with those premises. The inductive problem is just the inverse. Having given certiuu combinations of terms, we need to ascertain the propositions with which the combinations are consistent, and from which they may have proceeded. Now, if the reader contemi)lates the following combina- tions, ABC ohG aBC ahc, he will probably remember at once that they belong to the premises A = AB, B = BC (p. 92). If not, he will require a few trials before he meets with the right answer, and every trial will consist in assuming certain laws and observing whether the deduced results agree with the data. To test the facility with which he can solve this inductive problem, let him casually strike out any of the combina- tions of the fourth column of the Logical Alphabet, (p. 94), and say what laws the remaining combinations obey, observing that every one of the letter-terms and their negatives ought to appear in order to avoid self-contradic- tion in the premises (pp.74, in). Let him say, for instance, what laws are embodied in the combinations t 126 THE PRINCIPLES OF SCIENCE. [CHAi. ABC «BC Abe ahQ. The difficulty becomes much greater when more terms enter into the combinations. It would require some little examination to ascertain the complete conditions fulfilled in the combinations AC<5 ahCe aBC« abcK «BcfliE The reader may discover easily enough that the principal laws are C = «, and A = A^; but he would hardly discover without some trouble the remaining law, namely, that BD = BD«. The difficulties encountered in the inductive investigations of nature, are of an exactly similar kind. We seldom observe any law m uninternipted and undisguised opera- tion. The acuteness of Aristotle and the ancient Greeks did not enable them to detect that all terrestrial bodies tend to fall towards the centre of the earth. A few nights of observation might have convinced an astronomer viewing the solar system from its centre, that the planets travelled round the sun ; but the fact that our place of observation is one of the travelling planets, so conS)licates the apparent motions of the other bodies, that it required all the sagacity of Copernicus to prove the real simplicity of the planetary, system. It is the same throughout nature; the laws may be simple, but their combined effects are not simple, and we have no clue to guide us through their intricacies. " It is the glory of God," said Solomon, " to .conceal a thing, but the glory of a king to search it out" The laws of nature are the invaluable secrets which God has hidden, and it is the kingly pre- rogative of the philosopher to search them out by industry and sagacity. ^ Inductive Problems for Solution hy the Reader. In the first edition (vol ii. p. 370) I gave a logical problem involving six terms, and requested renders to discover the laws governing the combinations given. I received satisfactory replies from readers both in the United States and in England. I formed the combina- li^f ▼ II.] INDUCTION. 187 tions deductively from four laws of correction, but my correspondents found that three simpler laws, equivalent to the four more complex ones, were the best answer ; these laws ai-e as follows : a = ac, h = cd, d = Ef. In case other readers should like to test their skill in the inductive or inverse problem, I give below several series of combinations forming problems of graduated difficulty. Pmo.vlbm I. AbCD a b C D < A he D a b C d B L B e fiBO D a b e l> e A b aBe D a b e d E a BC a B e d a b Cd • Prublem IX, PkUtLBM II. Pkoblkm Tt. ABcDEP ABC ABCDE ABc D e P A b C D e/ Abe D E/ A b c D e / A b ed E F A b c d e F aBe D E P Be D e P a Bed £ F A b C a B C A B fi FkOtLEM III A B Cd« ABeD E A B e d e AbCD E aBCDE a B C d e abC D E abode ABC 6C D E P A b C (t ft C D e F a B C aBe PbOBLBM VII. b C D «/ a b e D « / a b e A b e D e aBC d E • 6 C d B ■ b e D E y abode ¥ Pkoilbm it Pboblbm X. ABCn PnOBtEM VIII. A be D ABC DeP a B e d ABCDE ABe D B/ « b C d ABC Di AbCDEF ABC de AbC De F AB e d e A b c D e F PROBLEM r. AbC DE aBC D E^ A bed B aB e D K/ Abode a bC D e F ABCD aB D < a b C d e F A BCd a B d e a b e D e / A Bed aB e D< abode/ Induction of Simple Identities, Many important laws of nature are expressible in the form of simple identities, and I can at once adduce them as examples to illustrate what I have said of the difficulty of the inverse process of induction. Two phenomena are conjoined. Thus all gravitating matter is exactly co- incident with all matter possessing inertia; where one ^-.LZ m THE PRINCIPLES OF SCIENCK [oBAf, II h property appears, the other likewise appears. All crystals of the cubical system, are all the crystals which do not doubly refract light. All exogenous plants are, with some exceptions, those which have two cotyledons or seed-leaves. A little reflection will show that there is no direct and infallible process by which such complete coincidences may be discovered. Natural objects are aggregates of many qualities, and any one of those qualities may prove to be in close connection with some others. If each of a numerous group of objects is endowed with a hundroc distinct physical or chemical qualities, there will be no less tlian I (lOO X 99) or 4950 pairs of qualities, which may be connected, and it will evidently be a matter of great intricacy and labour to ascertain exactly which qualities are connected by any simple law. One principal source of difficulty is that the finite powei-s of the human mind are not sufficient to compare by a single act any large group of objects with another large group. We cannot hold in the conscious possession of the mind at any one moment more than five or six different ideas. Hence we must tieat any more complex group by successive acts of attention. Tlie reader will perceive by an almost individual act of comparison that the words Roma and Mora contain the same letters. He may perhaps see at a glance whether the same is true of Causal and Casual^ and of Logica and Caligo. To assure himself that the letters in Astronomers make No more stars, that Serpens in akuleo is an anagram of Joannes Keplenis, or Great gun do us a sum an anagram of Au- gustus de Morgan, it will cei*tainly be necessary to break up the act of comparison into several successive acts. The process will acquire a double character, and will consist in ascertaining that each letter of the first group is among the letters of the second group, and vice versd, that each letter of the second is among those of the first group. In the same way we can only prove that two long lists of names are identical, by showing that each name in one list occurs in the other, and vice versd. This process of comparison really consists in establishing two partial identities, which are, as already shown (p. 58), equivalent in conjunction to one simple identity. We first ascertain the truth of the two propositions A = AB, VII.] INDUCTION. 1» B = AB, and we then rise by substitution to the single law A = B. ^ There is another process, it is true, by which we may get to exactly the same result ; for the two propositions A = AB, o = oi are also equivalent to the simple identity A = B. If then we can show that all objects included under A are included under B, and also that all objects not included under A are not included under B, our pur- pose is effected. By this process we should usually com- pare two lists if we are allowed to mark them. For each name in the first list we should strike off one in the second, and if, when the first list is exhausted, the second list is also exhausted, it follows that all names absent from the first must be absent from the second, and the coincidence must be complete. These two modes of proving an identity are so closely allied that it is doubtful how far we can detect any differ- ence in their powers and instances of application. The first method is perhaps more convenient when the pheno- mena to be compared are rare. Thus we prove that all the musical concords coincide with all the more simple numerical ratios, by showing that each concord arises from a simple ratio of undulations, and then showing that eack simple ratio gives rise to one of the concords. To examine all the possible cases of discord or complex ratio of undulation would be impossible. By a happy stroke of induction Sir John Herschel discovered that all crystals of quartz which cause the plane of polarization of light to rotate are precisely those crystals which have plagi- hedral faces, that is, oblique faces on the comers of the prism unsymmetrical with the ordinary faces. This singular relation would be proved by observing that all plagihedral crystals possessed the power of rotation, and Tfice versd all crystals possessing this power were plagi- hedral But it might at the same time be noticed that all ordinary crystals were devoid of the power. There is no reason why we should not detect any of the four pro- positions A = AB, B = AB, a = ah, b = ah, all of which- follow from A =» B (p. 115). Sometimes the terms of the identity may be singular objects ; thus we observe that diamond is a combustible gem, and being unable to discover any other that is, we affirm — )30 THE PRINCIPLES OF SCIENCE. [on AW Diamond = combustible gem. In a similar manner we ascertain that Mercury = metal liquid at ordinary temperatures, Substance of least density = substance of least atomic weight. Two or three objects may occasionally enter into the induction, as when we learn that Sodium -I* potassium = metal of less density than water, Venus •!• Mercury •!• Mars = major planet devoid of satellites. ' f IndvAiion. of Partial Identities, We found in the last section that the complete identity of two classes is almost always discovered not by direct observation of the fact, but by first establishing two partial identities. There are also a multitude of cases in which the partial identity of one class with another is the only relation to be discovered. Thus the most common of all inductive inferences consists in establishing the fact that all objects having the properties of A have also those of B, or that A = AB. To ascertain the truth of a pro- position of this kind it is merely necessary to assemble together, mentally or physically, all the objects included under A, and then observe whether B is present in each of them, or, which is the same, whether it would be im- possible to select from among them any not-B. Thus, if we mentally assemble together all the heavenly bodies which move with apparent rapidity, that is to say, the planets, we find that they all possess the property of not scintillating. We cannot analyse any vegetable substance without discovering that it contains carbon and hydrogen, but it is not true that all substances containing carbon and hydrogen are vegetable substances. The great mass of scientific truths consists of propo- sitions of this form A = AB. Thus in astronomy we learn that all the planets are spheroidal bodies ; that they all revolve in one direction round the sun ; that they all shine by reflected light; that they all obey the law of gravi- tation. But of course it is not to be asserted that all bodies obeying the law of gravitation, or shining by ▼II.] INDUCTION. 13) reflected light, or revolving in a particular direction, or being spheroidal in form, are planets. In other sciences we have immense numbers of propositions of the same form, as, for instance, all substances in becoming gaseous absorb heat ; all metals are elements ; they are' all good conductors of heat and electricity ; all the alkaline metals are monad elements; all foraminifera are marine organ- isms ; all parasitic animals are non-mammalian ; lightning never issues from stratous clouds; pumice never occurs where only Labrador felspar is present ; milkmaids do not suffer from small-pox ; and, in the works of Darwin, scientific importance may attach even to such an appa- rently trifling observation as that " white tom-cats having blue eyes are deaf." The process of inference by which all such truths are obtained may readily be exhibited in a precise symbolic form. We must have one premise specifying in a dis- junctive form all the possible individuals which belong to a class ; we resolve the class, in short, into its con- stituents. We then need a number of propositions, each of which affirms that one of the individuals possesses a certain property. Thus the premises must be of the forms A = B f. C i D .[. B=rBX C = CX + P + Q .: Q = QX. Now, if we substitute for eacli alternative of the first premise its description aa found among the succeedin*^ premises, we obtain " A = BX + CX + -H PX .|. QX or A = (B .|. C + .|. Q)X But for the aggregate of alternatives we may now substitute their equivalent as given in the firat premise, namely A, so that we get the required result : A = AX. We should have reached the same result if the first premise had been of the form A = AB .|. AC ^' + AQ. K 2 \> I *<! I« 132 THE PRINCIPLES OF S0IEN(2R. [OBAP. We can always prove a proposition, if we find it more convenient, by proving its equivalent, fo assert that all not-B*s are not-A's, is exactly the same as to assert that all A's are B's. Accordingly we may ascertain that A - AB by first ascertaining that b «- ab. If we observe, for instance, that all substances which are not solids are also not capable of double refraction, it follows necessarily that all double refracting substances are solids. We may convince our- selves that all electric substances are nonconductors of electricity, by reflecting that all good^ conductors do not, and in fact cannot, retain electric excitation. When we come to questions of probability it will be found desirable to prove, as far as possible, both the original proposition and its equivalent, as there is then an increased area of observation. The number of alternatives which may arise in the division of a class varies greatly, and may be any number from two upwards. Thus it is probable that every sub- stance is either magnetic or diamagnetic, and no substance can be both at the same time. The division then must be made in the form A = ABc + AbG. If now we can prove that all magnetic substances are capable of polarity, say B = BD, and also that all dia- magnetic substances are capable of polarity, C = CD, it follows by substitution that all substances are capable of polarity, or A = AD. We commonly divide the class sub- stance into the three subclasses, solid, liquid, and gas ; and if we can show that in each of these forms it obeys Carnot's thermodynamic law, it follows that all substances obey that law. Similarly we may show that all vertebrate animals possess red blood, if we can show separately that fish, reptiles, birds, marsupials, and mammals possess red blood, there being, as far as is known, only five principal subclasses of vertebrata. Our inductions will often be embarrassed by exceptions, real or apparent We might affirm that all gems are in- combustible were not diamonds undoubtedly combustible. Nothing seems more evident than that all the metals are opaque until we examine them in fine films, when gold and silver are found to be ti-ausparent. All plants absorb carbonic acid except certain fungi ; all the bodies of the VII.) INDUCTION. 133 planetary system, have a progressive motion from west to east, except the satellites of Uranus and Neptune. Even some of the profoundest laws of matter are not quite universal ; all solids expand by heat except india-nibber, and possibly a few other substances ; all liquids which have been tested expand by heat except water below 4° C. and fused bismuth; all gases have a coefficient of expansion increasing with the temperature, except hydrogen. In a later chapter I shall consider how such anomalous cases may be regarded and classified ; here we have only to expi-ess them in a consistent manner by our notation. Let us take the case of the transparency of metals, and D = iron E, F, &c. = copper, lead, &c. X = opaque. C D |. E, &c. Now evidently .)be, if assign the terms thus : — A s= meoal B = gold C = silver Our premises will be A = B B = Ba; C = Cte D=DX E«=EX, and so on for the rest of the metals. Abe = (D ^. E .|. F + and by substitution as before we shall obtain Abe = AbcX, or in words, "All metals not gold nor silver are opaque •/ at the same time we have A(B + C) = AB .|. AC = ABa; -I- AGx = A(B |. C)a;. or " Metals which are either gold or silver are not opaque." In some cases the problem of induction assumes a much higher degree of complexity. If we examine the properties of crystallized substances we may find some properties which are common to all, as cleavage or fracture in definite planes ; but it would soon become requisite to break up the class into several minor ones. We should divide crystals according to the seven accepted systems — and we should then find that crystals of each system possess many common properties. Thus crystals of the Regular or Cubical system expand equally by heat, conduct heat and electricity with uniform rapidity, and are of like elasticity in all directions; they have but one index of r 134 THE PRINCIPLES OF SCIENCE. [chap. r 1 1,1 ■ > refraction for light ; aud every facet i^ repeated in like relation to each of the three axes. Crystals of the system having one principal axis will be found to possess the various physical powers of conduction, refraction, elas- ticity, &c., uniformly in directions perpendicular to the principal axis ; in other directions their properties vary according to complicated laws. The remaining systems in which the crystals possess three unequal axes, or have inclined axes, exhibit still more complicated results, the effects of the crystal upon light, heat, electricity, &c., varying in all directions. But when we pursue induction into the intricacies of its application to nature we really enter upon the subject of classification, which we must take up again in a later part of this work. Solution of the Inverse or Inductive Problem, involving Tioo Classes. It is now plain that Induction consists in passing back from a series of combinations to the laws by which such combinations are governed. The natural law that all metals are conductors of electricity really means that in nature we find three classes of objects, namely — 1. Metals, conductors ; 2. Not-metals, conductors ; 3. Not-metals, not-conductors. It comes to the same thing if we say that it excludes the existence of the class, "metals not-conductors." In the same way every other law or group of laws will really mean the exclusion from existence of certain combinations of the things, circumstances or phenomena governed by . those laws. Now in logic, strictly speaking, we treat not the phenomena, nor the laws, but the general forms of the laws ; and a little consideration will show that for a finite number of things the possible number of forms or kinds of law governing them must also be finite. Using general terms, we know that A and B can be present or absent in four ways and no more — thus : AB, Ab, oB, ab; therefore every possible law which can exist concerning the relation of A and B must be marked by the exclusion •f one or more of the <».bove combinations. The number vn.] INDUCTION. 136 of possible laws then cannot exceed the number of selec- tions which we can make from these four combinations. Since each combination may be present or absent, the number of cases to be considered is 2 x 2 x 2 x 2, or sixteen ; and these cases are all shown in the following table, in which the sign o indicates absence or non-existence of the combination shown at tHe left-hand column in the same line, and the mark i its presence : — I 2 3 4 6 6 T • 9 10 • 11 12 • 18 14 • 15 16 • AB I m I I I I I I A6 t « 1 t t I I I aB I 1 I 1 I t I I ab I I I I t I I s ' 'Thus in colunm sixteen we find that all the conceivable combinations are present, which means that there are no special laws in existence in such a case, and that the combinations are governed only by the universal Laws of Identity and Difference. The example of metals and conductors of electricity would be represented by the twelfth column ; and every other mode in which two things or qualities might pi'esent themselves is shown in one or other of the columns. More than half the cases may indeed be at once rejected, because they involve the entire absence of a term or its negative. It has been shown to be a logical principle that every term must have its negative (p. 11 1), and when this is not the case, incon- sistency between the conditions of combination must exist. Thus if we laid down the two following propositions, " Graphite conducts electricity," and " Graphite does not conduct electricity," it would amount to asserting the impossibility of graphite existing at all ; or in general terms, A is B and A is not B result in destroying alto- gether the combinations containing A, a case shown in the fourth column of the above table. We therefore restrict our attention to those cases which may be represented in natural phenomena when at least two combinations are present, and which correspond to those columns of the 136 THE PRINCIPLES OP SCIENCE. [CHAF. l'! ■^T^gniV;.. ir I. table in which each of A, a, B, 6 appears. These cases are shown in the columns marked with an asterisk. We find that seven cases remain for examination, thus characterised — Four cases exhibiting three combinations, Two cases exhibiting two combinations. One case exhibiting four combinations.' It lias already been pointed out that a proposition of the form A = AB destroys one combination, AJ, so that this is the form of law applying to the twelfth column. But by changmg one or more of the terms in A = AB into its negative or by interchauging A and B, a and b, we obtain no less than eight different varieties of the one form ; thus— lathcase. 8th CMa. 15th ewe, t4thea8e. A = AB A = Aft a = aB a = ab b = ab B = aB b = Ab B = AB The reader of the preceding sections will see that each proposition in the lower line is logically equivalent to and 18 m fact the contrapositive of, that above it (p. ^i) Thus the propositions A - A6 and B = aB both give the same combinations, shown in the eighth column of the table and trial shows that the twelfth, eighth, fifteenth and fourteenth columns are thus accounted for. We come to this conclusion then— The general form of proposition A --AB admits of four logically distinct varieties, eack capable of expression in two modes. In two columns of the table, namely the seventh and tenth, we observe that two combinations are missing Now a simple identity A = B renders impossible both Ab and aB, accounting for the tenth case ; and if we change B into b the identity A = J accounts for the seventh case Ihere may indeed be two other varieties of the simple identity, namely a = & and a = B ; but it has already been shown repeatedly that these are equivalent respec tively to A = B and A = 6 (p. 115). As the sixteenth column has already been accounted for as governed by no special conditions, we come to the following general conclusion :— The laws governing the combinations of two terms must be capable of expression either in a partial Identity or a simple identity ; the partial identity is capable of only four logically distinct varieties, and the simple ^entity of two. Every logical relation between two terns vil] INDUCTION. 137 V must be expressed in one of these six forms of law, or must be logically equivalent to one of them. In short, we may conclude that in treating of partial and complete identity, we have exhaustively treated the modes in which two terms or classes of objects can be related. Of any two classes it can be said that one must either be included in the other, or must be identical with it, or a like relation must exist between one class and the negative of the other. We have thus completely solved the inverse logical problem concerning two terms.^ The Inverse Ijogical Problem involving Three Classes. No sooner do we introduce into the problem a third term C, than the investigation assumes a far more complex character, so that some readers may prefer to pass over this section. Three terms and their negatives may be combined, as we have frequently seen, in eight different combinations, and the effect of laws or logical conditions is to destroy any one or more of these combinations. Now we may make selections from eight things in 2" or 256 ways; so that we have no less than 256 different cases to treat, and the complete solution is at least fifty times as troublesome as with two terms. Many series of com- binations, indeed, are contradictory, as in the simpler problem, and may be passed over, the test of consistency being that each of the letters A, B, C, a, 6, c, shall appear somewhere in the series of combinations. My mode of solving the problem was as follows: — Having written out the whole of the 256 series of com- binations, I examined them separately and struck out such as did not fulfil the test of consistency. I then chose some form of proposition involving two or three terms, and varied it in every possible manner, both by the circular interchange of letters (A, B, C into B, C, A and then into C, A, B), and by the substitution for any one or more of the terms of the corresponding negative terms. _ • * The contents of this and the following section nearly correspond with those of a paper read before the Manchester Literary and Philosophical Society on December 26th, 187 1. See Proceedings of the Society, vol. xi. pp. 65—68, and Memoirs, Third Series, voL r. pp. 119-130. f 138 THE PRINCIPLES OP SCIENCE. [chap. VIL] INDUCTION. 139 w \l r For instance, the proposition AB = ABC can be first varied by circular interchange so as to give BC = BCA and then CA = CAB. Each of these three can then be thrown into eight varieties by negative change. Thus AB = ABC gives aB = aBC, Ab = A6C, AB = ABc, ab = ahC, and so on. Thus there may possibly exist no less than twenty- four varieties of the law having the general form AB = ABC, meaning th»nt whatever has the properties of A and B has those also of C. It by no means follows that some of the varieties may not be equivalent to others ; and trial shows, in fact, that AB = ABC is exactly the same in meaning as Ac = Abe or Be = Bca. Thus the law in question has but eight varieties of distinct logical mean- ing. I now ascertain by actual deductive reasoning which of the 256 series of combinations result from each of these distinct laws, and mark them off as soon as found. I then proceed to some other form of law, for instance A = ABC, meaning, that whatever has the qualities of A has those also of B and C. I find that it admits of twenty-four variations, all of which are found to be logically distinct ; the combinations being worked out, I am able to mark off twenty-four more of the list of 256 series. I proceed in this way to work out the results of every form of law which I can find or invent. If in the course of this work I obtain any series of combinations which had been pre- viously marked off, I learn at once that the law giving these combinations is logically equivalent to some law previously treated. It may be safely inferred that every variety of the apparently new law will coincide in meaning with some variety of the former expression of the same law. I have sufficiently verified this assumption in some cases, and have never found it lead to error. Thus as AB = ABC is equivalent to Ac = Abe, so we find that ab = ahC is equivalent to ac = ocB. Among the laws treated were the two A = AB and A — B which involve only two terms, because it may of course happen that among three things two only are in spedial logical relation, and the third independent; and the series of combinations representing such cases of re- lation are sure to occur in the complete enumeration. All single propositions which I could invent having been treated, pairs of propositions were next investigated. Thus wo have the relations, " All A's are B's and all B's are C's," of which the old logical syllogism is the development. We may also have " all A's are all B's, and all B*s are C's," or even "all A's are all B's, and all B's are all C's." All such premises admit of variations, greater or less in number, the logical distinctness of which can only be determined by trial in detail. Disjunctive propositions either singly or in pairs were also treated, but were often found to be equivalent to other propositions of a simpler form ; thus A = ABC -I- Abe is exactly the same in meaning as AB = AC. This mode of exhaustive trial bears some analogy to that ancient mathematical process called the Sieve of Eratosthenes. Having taken a long series of the natural numbers, Eratosthenes is said to have calculated out in succession all the multiples of every number, and to have marked them off, so that at last the prime numbers alone remained, and the factoi*s of every number were exhaustively discovered. My problem of 256 series of combinations is the logical analogue, the chief points of difference being that there is a limit to the number of cases, and that prime numbers have no analogue in logic, since every series of combinations corresponds to a law or group of conditions. But the analogy is perfect in the point that they are both inverse processes. There is no mode of ascertaining that a number is prime but by showing that it is not the product of any assignable factors. So there is no mode of ascertaining what laws are embodied in any series of combinations but trying exhaustively the laws which would give thenL Just as the results of Erato- sthenes' method have been worked out to a great extent and registered in tables for the convenience of other mathematicians, I have endeavoured to work out the inverse logical problem to the utmost extent which is at present practicable or ujefuL I have thus found that there are altogether fifteen con- ditions or series of conditions which may govern the com- binations of three terms, forming the premises of fifteen essentially different kinds of arguments. The following table contains a statement of these conditions, together with the numbers of combinations which are contradicted or destroyed by each, and the numbers of logically distinct 140 THE PKINCIPLES OF SCIENCE. [chap. variations of which the law is capable. There might be also added, as a sixteenth case, that case where no special logical condition exists, so that aU the eight combinations remain. !i !i(| I ^1 I ![!•,* •H Reference Number. riuposltione expressing tb« genenl type of ihe logical conditiona. 1 Nnmberofdi*. 1 tinct logical ▼ariatiooa f Namber of combinations eoutradictad by each. I. A = B 6 II. A = AB III. A = B, B = C la IV. A = B. B = BC 4 V, A = AB, B a BC •4 VI. A=:BC •4 VII. A = ABC •« VIII. AB = ABC t IX A = AB. aB = aBe X. A = ABC, ab m abC t XI. AB=sABC. ab=:abe XII. AB = AC 4 XIII. XIV. A =: BC 1- Afte A = BC j. be • XV. A -ABC. a=sBc.|. 6(7 • • There are sixty-three series of combinations derived from self-contradictory premises, which with 192, the snm of the numbers of distinct logical variations stated in the third column of the table, and with the one case where there are no conditions or laws at all, make up the whole conceivable number of 256 series. We learn from this table, for instance, that two pro- positions of the form A = AB, B = BC, which are such as constitute the premises of the old syllogism Barbara exclude as impossible four of the eight combinations in which three terms may be united, and that these proposi- tions are capable of taking twenty-four variations by tmns- positions of the terms or the introduction of natives This table then presents the results of a complete analysis of all the possible logical relations arising in the case of three terms, and the old syllogism forms but one out of fifteen typical forms. GeneraUy speaking, every form can h! T^^l^.^ '""^ apparently different propositions ; thus the fourth type A = B, B = BC may appeaV in the form A - Ai50, a = «J, or again m the form of three proposi- tions A = AB, B =BC, aB = aBc; but all these seHf premises yield identically the same series of combinations, ▼II.] INDUCTION. 141 and are therefore of equivalent logical meaning. The fifth type, or Barbara, can also be thrown into the equivalent forms A « ABC, aB = aBC and A = AC, B = A I- aBC. In other cases I have obtained the very same logical conditions in four modes of statements. As regards mere appearance and form of statement, the number of possible premises would be very great, and difficult to exhibit exhaustively. The most remarkable of all the types of logical condition is the fourteenth, namely, A = BC I- be. It is that which expresses the division of a genus into two doubly marked species, and might be illustrated by the example—" Com- ponent of the physical universe = matter, gravitating, or not-matter (ether), not-gravitating." It is capable of only two distinct logical variations, namely, A = BC •!• he and A = Be + 6C. By transposition or negative change of the letters we can indeed obtain six different expressions of each of these propositions ; but when their meanings are analysed, by working out the combinations, they are found to be logically equivalent to one or other of the above two. Thus the proposition A = BC •!• he can be written in any of the following five other modes, o = iC I- Be. B = CA .|. ca, J = cA I- Ca, C = AB .|. ah, c = aB + A*. I do not think it needful to publish at present the com- plete table of 193 series of combinations and the premises corresponding to each. Such a table enables us by mere inspection to learn the laws obeyed by any set of com- binations of three things, and is to logic what a table of factors and prime numbers is to the theory of numbers, or a table of integrals to the higher mathematics. The table already given (p. 140) would enable a person with but little labour to discover the law of any combinations. If there be seven combinations (one contradicted) the law must be of the eighth type, and the proper variety will be apparent. If there be six combinations (two contradicted), either the second, eleventh, or twelfth type applies, and a certain number of trials will disclose the proper type and variety. If there be but two combinations the law must be of the third type, and so on. The above investigations are complete as regards the possible logical relations of two or three terms. But ii M If I / ,<• 1^' ! \i\ iir^ 142 THE PRINCIPLES OF SCIENCE. [OHAF. when we attempt to apply the same kind of method to the relations of four or more terms, the labour becomes impracticably gi-eat Four terms give sixteen combinations compatible with the laws of thought, and the number of possible selections of combinations is no less than 2^* or 65,536. The following table shows the extraordinary manner in which the number of possible logical relations increases with the number of terms involved. Namber of terma. a S 4 I Namber of possible com< Innationi. 4 8 16 5 Nnmber of pouible selections of combinations corrasponding to consistent or inconsistent I(^cal relations. 16 156 ^5.53* . , 4,a94,967,a96 >»i44fi»744.073,709,55i,6i6 Some years of continuous labour would be required to ascertain the types of laws which may govern the com- binations of only four things, and but a small part of such laws would be exemplified or capable of practical appli- cation in science. The purely logical inverse problem, whereby we pass from combinations to their laws, is solved in the preceding pages, as far as it is likely to be for a long time to come ; and it is almost impossible that it should ever be carried more than a single step further. ^ ^ In the first edition, vol I p. 158, I stated' that I had not been able to discover any mode of calculating the number of cases in which inconsistency would be implied in the selection of combinations from the Logical Alphabet. The logical complexity of the problem appeared to be so great that the ordinary modes of calculating numbers of com- binations failed, in my opinion, to give any aid, and exhaustive examination of the combinations in detail seemed to be the only method applicable. This opinion, however, was mistaken, for both Mr. R. B. Hayward, of Harrow, and Mr. W. H. Brewer have calculated the numbers of inconsistent cases both for three and for four terms, without much difficulty. In the case of four terms they find that there are 1761 inconsistent selections and 63,774 consistent, which with one case wliere no I I ▼II.] INDUCTION. 143 condition exists, make up the total of 65,536 possible selections. The inconsistent cases are distributed in the manner shown in the following table ; — Number of Combi nations remaining. t • 3 4 5 6 7 8 9. to,&c. Namber of Inconsistent Cases. I x6 iia 35« 536 448 9*4 64 8 e 0, &e. When more than eight combinations of the Logical Alphabet (p. 94, column V.) remain unexcluded, there cannot be inconsistency. The whole numbers of ways of selecting o, 1,2, &c., combinations out of 16 are given in the 17th line of the Arithmetical Triangle given further on in the Chapter on Combinations and Permutations, the sum of the numbers in that line being 65,536. Professor Clifford on the Types of Compound Statement involving Four Classes. . In'the first edition (vol i. p. 163), I asserted that some years of labour would be required to ascertain even the precise number of types of law governing the combinations of four classes of things. Though I still believe that some years' labour would be required to work out the types themselves, it is clearly a mistake to suppose that the numbers of such types cannot be calculated with a reason- able amount of labour. Professor W. K Clifford having actually accomplished the task. His solution of the numerical problem involves the use of a complete new system of nomenclature and is far too intricate to be fully described here. I can only give a brief abstract of the results, and refer readers, who wish to follow out the reasoning, to the Proceedings of the Literary and Philo- sophical Society of Manchester, for the 9th January, 1877, voL xvi., p. 88, where Professor Clifford's paper is printed in full. By a simple statement Professor Clifford means the denial of the existence of any single combination or crossr 144 THE PRINCIPLES OF SCIENOB. [OHAF. 'i t». I i IN* • iill " division, of the classes, as in ABCD = o, or AbCd — a The denial of two or more such combinations is called a compound statement, and is further said to be twofold, threefold, &c., according to the number denied. Thus ABC = o is a twofold compound statement in regard to four classes, because it involves both ABCD = o and ABC<i = o. When two compound statements can be converted into one another by interchange of the classes, A, B, C, D, with each other or with their complementary classes, a, h, c, d, they are called similar, and all similar statements are said to belong to the same type. Two statements ai'e called complementary when they deny between them all the sixteen combinations without both denying any one ; or, which is the same thing, when each denies just those combinations which the other permits to exist It is obvious that when two statements are similar, the complementary statements will also be similar, and consequently for every type of n-fold statement, there is a complementary type of (i6 — 7t)-fold statement. It follows that we need only enumerate the types as far as the eighth order; for the types of more-than -eight-fold statement will already have been given as complementary to types of lower orders. One combination, ABCD, may be converted into another AhCd by interchanging one or more of the classes with the complementary classes. The number of such changes is called the distance, which in the above case is 2. In two similar compound statements the distances of the combinations denied must be the same ; but it does not follow that when all the distances are the same, the state- ments are similar. There is, however, ouly one example of two dissimilar statements having the same distances. When the distance is 4, the two combinations are said to be obverse to one another, and the statements denying them are called obverse statements, as in ABCD = o and ahcd = o or again AbCd = o and aBcD = o. When any one com- bination is given, called the origin, all the others may be grouped in respect of their relations to it as foUows : — Four are at distance one from it, and may be called proocimaies ; six are at distance two, and may be called mediates ; four are at distance three, and may be called ultimMes ; finally the obverse is at distance /our. ▼II.] INDUCTION. 146 Origin and four proximatea. oBCD ABCd— ABCD— AftCD A6cD Six mediates. a/>CD ABcD aBcD AhCd Obverse and four ultlmatea. Ahcd abcD—abcd — aBcd ABcd aBCd I abCd. It will be seen that the four proxiraates are respectively obverse to the four ultimates, and that the mediates form three pairs of obverses. Every proximate or ultimate is distant I and 3 respectively from such a pair of mediates. Aided by this system of nomenclature Professor Cliflford proceeds to an exhaustive enumeration of types, in which It IS impossible to follow him. The results are as follows — I -fold statements 2 3 4 5 6 / »> »f 8-fold statements » w » »» »» l> » >» t» «» I type 4 types 6 19 47 55 7S y* »» n » » 159 / Now as each seven-fold or less-than-seven-fold statement IS complementary to a nine-fold or more-than- nine-fold statement, it follows that the complete number of types will be 159 X 2 + 78 = 396. It appears then that the types of statement concernincr four classes are only about 26 times as numerous as those concerning three classes, fifteen in number, although the number of possible combinations is 256 times as great. Professor Clifford informs me that the knowledge of the possible groupings of subdivisions of classes which he obtained by this inquiry has been of service to him in some applications of hyper-elliptic functions to which he lias subsequently been led. Professor Cayley has since expressed his opinion that this line of investigation should be followed out, owing to the bearing of the theory of compound combinations upon the higher geometry.^ It seems likely that many unexpected points of connection fitK ^*'^«^*«^«y '^ Manchester Literary and Philosophical Soci^m, oth Febniaiy, 1877, vol. xvl, p. 1 13. 146 THE PRINCIPLES OF SCIENCE. [CHAF. ▼il] INDUCTION. !'i f'' I will in time be disclosed between the sciences of logic and mathematics. Distinction between Perfect and Imperfect IndiLction. We cannot proceed with advantage befoi-e noticing the extreme difference which exists between cases of perfect and those of imperfect induction. We call an induction perfect when all the objects or events which can possibly come under the class treated have been examined. But in the majority of cases it is impossible to collect together, or in any way to investigate, the properties of all portions of a substance or of all the individuals of a race. Tlie number of objects would often be practically infinite, and the greater part of them might be beyond our reach, in the interior of the earth, or in the most distant parts of the Universe. In all such cases induction is imperfeety and is affected by more or less uncertainty. As some writers have fallen into much error concerning the func- tions and relative importance of these two branches of reasoning, I shall have to point out that — 1. Perfect Induction is a process absolutely requisite, l)oth in the performance of imperfect induction and in the treatment of large bodies of facts of wliich our knowledge is complete. 2. Imperfect Induction is founded on Perfect Induction, but involves another process of inference of a widely different character. It is certain that if I can draw any inference at all concerning objects not examined, it must be done on the data aflbrded by the objects which have been examined. If I judge that a distant star obeys the law of gravity, it must be because all other material objects sufficiently known to me obey that law. If I venture to assert that all ruminant animals have cloven hoofs, it is because all ruminant animals which have come under my notice have cloven hoofs. On the other hand, I cannot safely say that all cryptogamous plants possess a purely cellular structure, because some cryptogamous plants, which have been examined by botanists, have a partially vascular structure. The probability that a new cryptogam will be cellular only can be estimated, if at all, on the ground of 147 the comparative numbers of known cryptogams which are and are not cellular. Thus the first step in every induction will consist in accurately summing up the number of instances of a particular phenomenon which have fallen under our observation. Adams and Leverrier, for instance, must have inferred that the undiscovered planet Neptune would obey Bode's law, because all the planets known at that time obeyed it. On what principles the passage from the known to the apparently unknown is warranted, must be carefully discussed in the next sec- tion, and in various parts of this work. It would be a great mistake, however, to suppose that Perfect Induction is in itself useless. Even when the enumeration of objects belonging to any class is complete, and admits of no inference to unexamined objects, the statement of our knowledge in a general proposition is a process of so much importance that we may consider it necessary. In many cases we may render our investiga- tions exhaustive ; all the teeth or bones of an animal ; all the cells in a minute vegetable organ ; all the caves in a mountain side ; all the strata in a geological section ; all the coins in a newly found hoard, may be so completely scrutinized that we may make some general assertion concerning them without fear of mistake. Every bone might be proved to cont^ain phosphate of lime ; every cell to enclose a nucleus ; every cave to hide remains of extinct animals ; every stratum to exhibit signs of marine origin ; every coin to be of Roman manufacture. These are cases where our investigation is limited to a definite portion of matter, or a definite area on the earth's surface. There is another class of cases where induction is naturally and necessarily limited to a definite number of alternatives. Of the regular solids we can say without the least doubt that no one has more than twenty faces, thirty edges, and twenty comers ; for by the principles of geometry we learn that there cannot exist more than five regular solids, of each of which we easily observe that the above statements are true. In the theory of numbers, an endless variety of perfect inductions might be made ; we can show that no number less than sixty possesses so many divisors, and the like is true of 360 ; for it does not require a great amount of labour to ascertain and count all the divisors L 2 i m 148 THE PRINCIPLES OF SCIENCE. [chap. of numbers up to sixty or 360. I can assert that between 60,041 and 60,077 no prime number occurs, because the exhaustive examination of those who have constructed tables of prime numbers proves it to be so. In matters of human appointment or history, we can frequently have a complete limitation of the number of instances to be included in an induction. We might show that the propositions of the third book of Euclid treat only of circles ; that no part of the works of Galen mentions the fourth figure of the syllogism ; that none of the other kings of England reigned so long as George III.; that Magna Charta has not been repealed by any subsequent statute ; that the price of corn in England has never been so high since 1847 as it was in that year; that the price of the English funds has never been lower than it was on the 23rd of January, 1798, when it fell to 47 J. It has been urged against this process of Perfect Induc- tion that it gives no new information, and is merely a summing up in a brief form of a multitude of particulars. But mere abbreviation of mental labour is one of the most important aids we can enjoy in the acquisition of knowledge. The powers of the human mind ai*e so limited that multi- plicity of detail is alone sufficient to prevent its progress in many directions. Thought would be practically impos- sible if every separate fact had to be separately thought and treated. Economy of mental power may be considered one of the main conditions on which our elevated intellectual position depends. Mathematical processes are for the most part but abbreviations of the simpler acts of addition and subtraction. The invention of logarithms was one of the most striking additions ever made to human power : yet it was a mere abbreviation of operations which could have been done before had a sufficient amount of labour been available. Similar additions to our power will, it is hoped, be made from time to time ; for the number of mathematical problems hitherto solved is but an indefinitely small fraction of those which await solution, because the labour they have hitherto demanded renders them impracticable. So it is throughout all regions of thought. The amount of our knowledge depends upon our power of bringing it within practicable compass. Unless we arrange and classify facts and condense them into general truths, they 7U.] INDUCTION. 149 soon surpass our powers of memory, and serve but to confuse. Hence Perfect Induction, even as a process of abbreviation, is absolutely essential to any high degree of mental achievement Transition frovi Perfect to Imperfect Induction. It is a question of profound difficulty on what grounds we are warranted in inferring the future from the present, or the nature of undiscovered objects from those which we liave examined with our senses. We pass from Perfect to Imperfect Induction when once we allow our conclusion to apply, at all events apparently, beyond the data on which it was founded. In making such a step we seem to gain a net addition to our knowledge ; for we learn the nature of what was unknown. We reap where we have never sown. We appear to possess the divine power of creating know- ledge, and reaching with our mental arms far beyond the sphere of our own observation. I shall have, indeed, U\ point out certain methods of reasoning in which we dt. pass altogether beyond the sphere of the senses, and acquire accurate knowledge which observation could never have given ; but it is not imperfect induction that accomplishes such a task. Of imperfect induction itself, I venture to assert thatdt never makes any real addition to our knowledge, in the meaning of the expression some- times accepted. As in other cases of inference, it merely unfolds the information contained in past observations; it merely renders explicit v.'hat was implicit in previous experience. It transmutes, but certainly does not create knowledge. There is no fact which I shall more constantly keep before the reader's mind in the following pages than that the results of imperfect induction, however well authen- ticated and verified, are never more than probable. Wo never can be sure that the future will be as the present. We hang ever upon the will of the Creator: and it is only so far as He lias created two things alike, or maintains the framework of the world unchanged from moment to moment, that our most careful inferences can be fulfilled. All predictions, all inferences which reach beyond their data, are purely hypothetical, and proceed on the assump- M I' i III 160 THE PRINCIPLES OP SCIENCE. [cnAp. tion that new events will conform to the conditions detected in our observation of past events. No experience of finite duration can give an exhaustive knowledge of the forces which are in operation. There is thus a double uncertainty • even supposing the Universe as a whole to proceed un- changed, we do not really know tlie Universe as a wliole We know only a point in its infinite extent, and a moment m Its infinite duration. We cannot bo sure, then, that our observations have not escaped some fact, which will cause the future to be apparently different from the past • nor can we be sure that the future really will be the outcome of the past. We proceed then in all our inferences to unexamined objects and times on the assumptions 1. That our past observation gives us a complete know- ledge of what exists. 2. That the conditions of things which did exist will continue to be the conditions which will exist. We shall often need to illustrate the character of our knowledge of nature by the simile of a ballot-box, so often employed by mathematical writera in the theory of proba- bility. Nature is to us like an infinite ballot-box the contents of which are being continually drawn, ball after ball, and exhibited to us. Science is but the careful observation of the succession in which balls of various character pi-eseiit themselves; we register the combina- tions, notice those which seem to be excluded from occur- rence, and from the proportional frequency of those which appear we infer the probable character of future drawings But under such circumstances certainty of prediction depends on two conditions : — I. Thai we acquire a perfect knowledge of the com- parative numbei-s of balls of each kind within the box. 2^ That the contents of the ballot-box remain unchanrred Of the latter assumption, or rather that conceniinc? the constitution of the world which it illustrates, the lo«?ician or physicist can have nothing to say. As the Creatfon of the Universe IS necessarily an act passing aU experience and all conception, so any change in that Universe or it may be, a termination of it, must likewise be infinitely be- yond the bounds of our mental faculties. No 8cien(4 no ▼II.] INDUCTION. 161 reasoning upon the subject, can have any validity; for without experience we are without the basis and materials of knowledge. It is the fundamental postulate accordingly of all inference concerning the future, that there shall be no arbitrary change in the subject of inference ; of the pro- bability or improbability of such a change I conceive that our faculties can give no estimate. The other condition of inductive inference — that we acquire an approximately complete knowledge of the com- binations in which events do occur, is in some degree within our power. There are branches of science in which phenomena seem to be governed by conditions of a most fixed and general character. We have ground in such cases for believing that the future occurrence of such phenomena can be calculated and predicted. But the whole question now becomes one of probability and im- probability. We seem to leave the region of logic to enter one in which the number of events is the ground of in- ference. We do not really leave the region of logic ; we only leave that where certainty, affirmative or negative, is the i-esult, and the agieement or disagreement of qualities the means of inference. For the future, number and quantity will commonly enter into our processes of reason- ing ; but then I hold that number and quantity are but portions of the great logical domain. I venture to assert that number is wholly logical, both in its fundamental nature and in its developments. Quantity in all its forms is but a development of number. That which is mathe- matical is nut the less logical ; if anything it is more logical, in the sense that it presents logical results in a higher degree of complexity and variety. Before proceeding then from Perfect to Imperfect In- duction I must devote a portion of this work to treating the logical conditions of number. I shall then employ number to estimate the variety of combinations in which natural phenomena may present themselves, and the pro- bability or improbability of their occurrence under definite circumstances. It is in later parts of the work that I must endeavour to establish the notions which I have set forth upon the subject of Imperfect Induction, as applied in the investigation of Nature, which notions may be thus briefly stated : — 1 1 IM THE PRINCIPLES OF SCIENCE. [chap. tii. 1. Imperfect Induction entirely rests upon Perfect In- duction for its materials. 2. The logical process by which we seem to pass directly from examined to unexamined cases consists in an inverse application of deductive inference, so that all reasoning may be said to be either directly or inversely deductive. 3. The result is always of a hypothetical character, and is never more than probable. 4. No net addition is ever made to our knowledge by reasoning ; what we know of future evente or un- examined objects is only the unfolded contents of our previous knowledge, and it becomes less pro- bable as it is more boldly extended to remote case& \ IM- BOOK II. NUMBER, VARIETY, AND PROBABILITY. CHAPTER VIII. PRINCIPLES OP NUMBER. Not without reason did Pythagoras represent the world as ruled by number. Into almost all our acts of thought number enters, and in proportion as we can define numeri- caUy we enjoy exact and useful knowledge of the Universe The science of numbers, too, has hitherto presented the widest and most practicable training in logic. So free and energetic has been the study of mathematical forms, com- pared with the forms of logic, that mathematicians have passed far m advance of pure logicians. Occasionally, in recent times, they have condescended to apply their algebraic instrument to a reflex treatment of the primary logical science. It is thus that we owe to profound mathe- maticians, such as John Herschel, WheweU, De Morgan, or Boole, the regeneration of logic in the present century ' I entertain no doubt that it is in maintaining a close alliance with quantitative reasoning that we must look for further progress in our comprehension of quaUtative inference I cannot assent, indeed, to the common notion that certainty begins and ends with numerical determination. JNothmg IS more certain than logical truth. The law* of Identity and difference are the tests of all that is certain i! I !. ! I i^ 154 THE PRINCIPLES OF SCIENCE. [chap. throughout the range of thought, and mathematical reason- ing is cogent only when it couforms to these conditions, of which logic is the first development And if it be erroneous to suppose that all certainty is mathematical, it is equally an error to imagine that all which is mathe- matical is certain. Many processes of mathematical reasoning are of most doubtful validity. There are points of mathematical doctrine which must long remain matter of opinion ; for instance, the best form of the definition and axiom concerning parallel lines, or the true nature of a limit. In the use of symbolic reasoning questions occur on which the best mathematicians may differ, as Bernoulli and Leibnitz differed irreconcileably concerning the exis- tence of the logarithms of negative quantities.^ In fact we no sooner leave the simple logical conditions of number, than we find ourselves involved in a mazy and mysterious science of symbols. Mathematical science enjoys no monopoly, and not even a supremacy, in certainty of results. It is the boundless extent and variety of quantitative questions that delights the mathematical student When simple logic can give but a bare answer Yes or No, the algebraist raises a score of subtle questions, and brings out a crowd of curious results. The flower and the fmit, all that is attractive and delightful, fall to the share of tlie mathematician, who too often despises the plain but necessary stem from which all has arisen. In no region of thought can a reasoner cast himself free from the prior conditions of logical cor- rectness. The mathematician is only strong and true as long as he is logical, and if number rules the world, it is logic which rules number. Nearly all writers have hitherto been strangely content to look upon numerical reasoning as something apart from logical inference. A long divorce has existed between quality and quantity, and it has not been uncommon to treat them as contrasted in nature and restricted to independent branches of thought For my own part, I believe that all the sciences meet somewhere. No part of knowledge can stand wholly disconnected from other parts of the universe of thought ; it is incredible, above all, that ' MoQtucla, Uistoire dc* MaUUmaiiquet, vol. iii. p. 373. VIII.] PRINCIPLES OF NUMBER. 166 the two great branches of abstract science, interlacing and co-operating in every discourse, should rest upon totally distinct foundations. I assume that a connection exists, and care only to inquire. What is its nature ? Does the science of quantity rest upon that of quality; or, vice versd, does the science of quality rest upon that of quantity? There might conceivably be a third view, that tbey both rest upon some still deeper set of prin- ciples. It is generally suj>posed that Boole adopted the second view, and treated logic as an application of algebra, a special case of analytical reasoning which admits only two quantities, unity and zero. It is not easy to ascertain clearly which of these views really was accepted by Boole. In his interesting biographical sketch of Boole,^ the Eev. K. Harley protests against the statement that Boole's logical calculus imported the conditions of number and quantity into logic. He says : « Logic is never identified or confounded with mathematics; the two systems of thought are kept perfectly distinct, each being subject to its own laws and conditions. The symbols are the same for both systems, but they have not the same intei-pre- tatiou." The Eev. J. Venn, again, in his review of Boole's logical system,2 holds that Boole's processes are at bottom logical, not mathematical, though stated in a highly gener- alized fonn and with a mathematical dress. But it is quite likely that readers of Boole should be misled. Not only have his logical works an entirely mathematical appearance, but I find on p. 12 of his Laws of TJioufjht the following unequivocal statement: "That logic, as a science, is susceptible of very wide applications is admitted; but it is equally certain that its ultimate forms and processes are mathematical" A few lines below he adds, " It is not of the essence of mathematics to be conversant with the ideas of number and quantity." The solution of the difficulty is that Boole used the terna mathematics in a wider sense than that usually attributed to it He pi-obably adopted the third view, so that his mathematical Laws of Thought are the common » British Quarterly Review, No. Ixxxvii, July 1866. ' Mind; October 1876, vol. i. p. 484. 186 THE PRINCIPLES OF SCIENCK [chap. i basis both of logic and of quantitative mathematics. But I do not care to pursue the subject because I think that in either case Boole was wrong. In my opinion logic is the superior science, the general basis of mathematics as well as of all other sciences. Number is but logical dis- crimination, and algebra a highly developed logic. Thus it is easy to understand the deep analogy which Boole pointed out between the fonns of algebraic and logical deduction. Logic resembles algebra as the mould resembles that which is cast in it Boole mistook the cast for the mould. Considering that logic imposes its own laws upon every branch of mathematical science, it is no wonder that we constantly meet with the traces of logical laws in mathematical processes. The Nature of Number. Number is but another name for diversity. Exact iden- tity is unity, and with difference arises plurality. An abstract notion, as was pointed out (p. 28), possesses a certain oneness. The quality of fustice, for instance, is one and the same in whatever just acts it is manifested. In justice itself there are no marks of difference by which to discriminate justice from justice. But one just act can be discriminated from another just act by circumstances of time and place, and we can count many acts thus discri- minated each from each. In like manner pure gold is simply pure gold, and is so far one and the same through- out. But besides its intrinsic qualities, gold occupies space and muse have shape and size. Poi-tions of gold are always mutually exclusive and capable of discrimina- tion, in respect that they must be each without the other. Hence they may be numbered. Plurality arises when and only when we detect differ- ence. For instance, in counting a number of gold coins I must count each coin once, and not more than once. Let C denote a coin, and the mark above it the order of counting. Then I must count the coins C + C" + C" + C" + If I were to count them as follows C + C + C"' + C" + CT'' + . . ., I should make the tliird coin into two, and should imply ▼III.] PRINCIPLES OF NUMBER. 157 the existence of difference where there is no difference.* C" and C' are but the names of one coin named twice over. But according to one of the conditions of logical symbols, which I have called the Law of Unity (p. 72), the same name repeated has no effect, and A + A = A. We must apply the Law of Unity, and must reduce all identical alternatives before we can count with certainty and use the processes of numerical calculation. Identical alternatives are harmless in logic, but are wholly inad- missible in number. Thus logical science ascertains the nature of the mathematical unit, and the definition may be given in these terms — A unit is any object of thought which can he discriminated from every other object treated as a unit in the same problem. It has often been said that units are unfts in respect of being perfectly similar to each other ; but though they may be perfectly similar in some respects, they must be different in at least one point, otherwise they would be incapable of plurality. If three coins were so similar that they occupied the same space at the same time, they would not be three coins, but one coin. It is a property of space that every point is discriminable from every other point, and in time every moment is necessarily distinct from any other moment before or after. Hence we frequently count in space or time, and Locke, with some other philosophers, has held that number arises from repetition in time. Beats of a pendulum may be so perfectly similar that we can discover no difference except that one beat is before and another after. Time alone is here the ground of difference and is a sufficient foundation for the discrimination of plurality ; but it is by no means the only foundation. Three coins are three coins, whether we count them successively or regard them all simul- taneously. In many cases neither time nor space is the ground of difference, but pure quality alone enters. We can discriminate the weight, inertia, and hardness of gold as three qualities, though none of these is before nor after the other, neither in space nor tima Every means of discrimination may be a source of plurality. * Pwr€ Logic, Appendix, p. 82, \ 192 158 THE PRINCIPLES OP SCIENCE. [CWAP. VIII.] PRINCIPLES OP NUMBER. 159 'V (■ mi. « Our logical notation may be used to express the rise of number. The symbol A stands for one thing or one class, and in itself must be regarded as a unit, because no difference is specified. But the combinations AB and Ah are necesssarily two, because they cannot logically coalesce, and there is a mark B which distinguishes one from the other. A logical definition of the number four is given in the combinations ABC, ABc, AhC, Ahc, where there is a double difference. As Puck says — " Yet but three ? Come one more ; Two of both kinds makes up four." I conceive that all numl)ers might be represented as arismg out of the combinations of the logical Alphabet, more or less of each series being struck out by various logical conditions. The number three, for instance, arises from the condition that A must be either B or C, so that the combinations are ABC, ABc, AbC. 0/ Numerical Abstraction. Tliere will now be little difficulty in forming a clear notion of the nature of numerical abstraction. It consists in abstracting the character of the difference from which plurality anses, retaining merely the fact. When I speak ot three men I need not at once specify the marks by which each may be known from each. Those marks must exist if they are really three men and not one and the same and m speaking of them as many I imply the existence of the requisite differences. Abstract number, then, is iJie empty form of difference ; the abstract number three asserts the ex- istence of marks without specifying their kind. Numerical abstraction is thus seen to be a dif- ferent process from logical abstraction (p. 27), for in the latter process we drop out of notice the very existence of difference and pluraUty. In forming the abstract notion hardriess we ignore entirely the diverse circumstances in which the quality may appear. It is the concrete notion three hard objects, which asserts the existence of hardness along with sufficient other undefined qualities, to mark out three such objects. Numerical thought is indeed closely interwoven with logical thought. We cannot use a con Crete term in the plural, as men, without implying that there are marks of difference. But when we use an abstract term, we deal with unity. The origin of the great generality of number is now apparent. Three sounds differ from three colours, or three riders from three horses ; but they agree in respect of the variety of marks by which they can be discriminated. The symbols 1+1+ 1 are thus the empty marks asserting the existence of discrimination. But in dropping out of sight the character of the differences we give rise to new agreements on which mathematical reasoning is founded. Numerical abstraction is so far from being incompatible with logical abstraction that it is tlie origin of our widest acts of generalization. Concrete and Abstract Numher. The common distinction between concrete and abstract number can now be easily stated. In proportion as we specify the logical characters of the things numbered, we render them concrete. In the abstract number thru there is no statement of the points in which the three objects agree ; but in three coins, three men, or three Jwrses, not only are the objects numbered but their nature is re- stricted. Concrete number thus implies^ the same con- sciousness of difference as abstract number, but it is mingled with a groundwork of similarity expressed in the logical terms. There is identity so far as logical terms enter ; difference so far as the terms are merely numerical. The reason of the important I^w of Homogeneity will now be apparent. This law asserts that in every arith- metical calculation the logical nature of the things num- bered must remain unaltered. The specified logical agreement of the things must not be affected by the un- specified numericsd differences. A calculation would be palpably absurd which, after commencing with length, gave a result in hours. It is equally absurd, in a purely arithmetical point of view, to deduce areas from the calculation of lengths, masses from the combination of volume and density, or momenta from mass and velocity. It must remain for subsequent consideration to decide in what sense we may truly say that two linear feet multi- .V. 160 THE PRINCIPLES OF SCIENCE. [chap nil.] PRINCIPLES OF NUMBER. 161 plied by two linear feet give four superficial feet ; arith- metically it is absurd, because there is a change of unit. As a general rule we treat in each calculation only objects of one natura We do not, and cannot properly add, in the same sum yards of cloth and pounds of sugar We cannot even conceive the result of adding area to velocity, or length to density, or weight to value. The units added must have a basis of homogeneity, or must be reducible to some common denominator. Nevertheless it is possible, and in fact common, to treat in one complex calculation the most heterogeneous quantities, on the condition that each kind of object is kept distinct, and treated numerically only in conjunction with its own kind. Different units, so far as their logical differences are speci- fied, must never be substituted one for the other. Chemists continually use equations which assert the equivalence of groups of atoms. Ordinary fermentation is represented by the fornmla C* H" 0* = 20* H« O + 200«. Three kinds of units, the atoms respectively of carbon, hydrogen, and oxygen, are here intermingled, but there is really a separate equation in regard to each kind. Mathe- maticians also employ compound equations of the same kind ; for in, a + J v/ - I = c 4- ^ v/ - I, it is impossible by ordinary addition to add atohy/— i. Hence we really have the separate equations a = b, and c ij — i = d J — I. Similarly an equation between two quaternions is equivalent to four equations between ordinary quantities, whence indeed the name quaternion. Analogy of Logical and Numtrical Terms. If my assertion is correct that number arises out of logical conditions, we ought to find number obeying all the laws of logic. It is almost superfluous to point out that this is the case with the fundamental laws of identity and difference, and it only remains to show that mathematical symbols do really obey the special conditions of logical symbols which were formerly pointed out (p. 32). Thus the Law of Oommutativeness, is equally true of quality and quantity. As in logic we have AB = BA, 80 in mathematics it is familiarly known that 2x3 = 3x2, or X X If =^y X X. The properties of space are as indifferent in multiplication as we found them m pure logical thought. Similarly, as in logic triangle or square = square or triangle or generally A + B = B .|. A, • 80 in quantity 2 + 3 = 3 + 2' or generally x -^ y = y + x. The symbol f. is not identical with +, but it is thus far analogous. How far now, is it true that mathematical symbols obey the Law of Simplicity expressed in the form AA = A, or the example Round round = round ? Apparently there are but two numbers which obey this law ; for it is certain that "^ XXX ^s v is true only in the two cases when a; = i, or a; = o In reality all numbers obey the law, for 2 x 2 ='2 is not really analogous to AA = A. According to the definition of a unit already given, each unit is discriminated from each other m the same problem, so that in 2' x 2" the first two involves a different discrimination from the second two. I get four kinds of things, for instance, if I first dis- criminate •' heavy and light" and then "cubical and spherical, for we now have the foUowing classes- heavy, cubical light, cubical heavy, spherical. light, spherical But suppose that my two classes are in both cases dis- weT^ve ^^™^ difference of light and heavy then heavy heavy = heavy, heavy light = o, light heavy = o, Hght light = light Ihus, Oieavy or light) x (heavy or light) = (heavy or light). In short, twice two ts two unless we take care that the second two has a different meaning from the first. But l.nH ^'"^"^I circumstances logical terms give the like result, and it is not true that A'A" = A', when A" is different in meaning from A'. ^mmm 102 THE PRINCIPLES OF SCIENCE. [OBAP. ▼!«.]• PRINCIPLES OF NUMBER. lea ,'V »r ■A f In a similar manner it may be shown that the Law of Unity A i A = A. holds true alike of logical and mathematical terms. It is absurd indeed to say that except in the one case when x = absolute zero. But this contradiction a; + a; = a; arises from the fact that we have already defined the units in one x as differing from those in the other. Under such circumstances tlie Law of Unity does not apply. For if in A' + A-' -A' we mean that A" is in any way different from A' the assertion of identity is evidently false. The contrast then which seems to exist between logical and mathematical symbols is only apparent It is because the Laws of Simplicity and Unity must always be observed in the operation of counting that those laws s6em no further to apply. This is the understood condition under which we use all numerical symbols. Whenever I wnte the symbol 5 I really mean I + I 4 I + I + I, . . and it is perfectly understood that each of these units is distiuct from each other. If requisite I might mark them thus "'+ !'"'+ i'"". i'+ r + I Were this not the case and were the units really I' + I" + I" + I'" + I"", the Law of Unity would, as before remarked, apply, and l" 4- I" = I". Mathematical symbols then obey all the laws of logical symbols, but two of these laws seem to be inapplicable simply because they are presupposed in the definition of the mathematical unit. Logic thus lays down the con- ditions of number, and the science of arithmetic developed as it is into all the wondrous branches of mathematical calculus is but an outgrowth of logical discrimination. Principle of Mathematical Inference. The universal principle of all reasoning, as I have asserted, is that which allows us to substitute like for like. I have now to point out how in the mathematical sciences this principle is involved in each step of reasoning. It is m these sciences indeed that we meet with the clearest cases of substitution, and it is the simplicity with which the principle can be applied which probably led to the compamtively early perfection of the sciences of geometry and arithmetic. Euclid, and the Greek mathematicians from the first, recognised equality as the fundamental relation of quantitative thought, but Aristotle rejected the exactly analogous, but far more general relation of identity and thus crippled the formal science of logic as it has descended to the present day. ^ Geometrical reasoning starts from the axiom that "things equal to the same thing are equal to each other " Two cquahties enable us to infer a third equality ; and this IS true not only of lines and angles, but of areas, volumes, numbers, intervals of time, forces, velocities, degrees of intensity, or, in short, anything which is capable of being equal or unequal. Two stars equally bright with the same star must be equally bright with each other, and two forces equally intense with a third force are equally intense with each other. It is remarkable that Euclid has not explicitly stated two other axioms, the truth of which is necessarily ^^ • l^* "^^^ second axiom should be that " Two things of which one is equal and the other unequal to a third com- mon thing, are unequal to each other." An equality and inequality, in short, give an inequality, and this is equaUy true with the first axiom of all kinds of quantity. If Venus, for instance, agrees with Mars in density, but Mars differs from Jupiter, then Venus differs from Jupiter. A third axiom must exist to the effect that " Things unequal to the same thing may or may not be equal to each other. Two inequalities give no ground of inference wJcat- wer. If we only know, for instance, that Mercury and Jupiter differ m density from Mars, we cannot say whether or not they agree between themselves. As a fact they do not agree ; but Venus and Mars on the other hand both differ from Jupiter and yet closely agree with each other. Ihe force of the axioms can be most clearly illustrated by drawing equal and unequal lines.* ««1 '?^^^«'^ iwwTW tn Logui (Macmillan), p. 123. It is pointed w^ J^ ^i- r^w ^i^^l* ^'^"^ ^^'^""^ thit'^the views her? gfv^ were partially stated by Leibnitx, ^ M 2 164 THE PRINCIPLES OF SCIENOE. [#HAP. ▼III.] PRINCIPLES OF NUMBER. 165 -3| w ' I The general conclusion then must be that where there is equality there may be inference, but where there is not equality there cannot be inference. A plain induction will lead us to believe that eqicality is tlu condition of inference concerning quantity. All the three axioms may in fact be summed up in one, to the effect, that "tn whatever relation one quantity stands to another, it stands in the same relation to tJie equal of that other" The active power is always the substitution of equals, and it is an accident that in a pair of equalities we can make the substitution in two ways. From a = 6 = c we can infer a = c, either by substituting in a = b the value of 6 as given in b = c, or else by substituting in J = c the value of b as given in a = b. In a = 6 *« rf we can make but the one substitution of a for J. In «-'/-' </ we can make no substitution and get no inference. In mathematics the relations in which terms may stand to each other are for more varied than in pure logic, yet our principle of substitution always holds true. We may say in the most general manner that In whatever relation one quantity stands to another, it stands in the same relation to the equal of that other. In this axiom we sum up a number of axioms which have been stated in more or less detail by algebraists. Tlius, " If equal quantities be added to equal quantities, the sums will be equal." To explain this, let Now a -\- c, whatever it means, must be identical with itself, so that a + c = a •\' e. In one side of this equation substitute for the quantities their equivalents, and we have the axiom proved a + c = 6 + (£. The similar axiom concerning subtraction is equally evi- dent, for whatever a — c may mean it is equal to a — c, ' and therefore by substitution Xx> b — d. Again, " if equal quantities be multiplied by the same op equal quantities, the products will be equal" For evidently a4i = ac^ and if for c in one side we substitute its equal d, we have ac = ad, and a second similar substitution gives us ac = hd. We might prove a like axiom concerning division in afi exactly similar manner. I might even extend the list of axioms and say that " Equal powers of equal numbers arc equal." For certainly, whatever ay ax a may mean, it is equal to a x a x « ; hence bv our usual substitution it IS equal to bxbxb. The same will be true of roots of numbers and IJa = *Jb provided that the roots are so taken that the root of a shall really be related to a as the root of b is to b. The ambiguity of meaning of an an operation thus fails in any way to shake the univ'ersality of the principle. We may go further and assert that, not only the above common relations, but all other known or conceivable mathematical relations obey the same prin- ciple. Let Qa denote in the most general manner that we do something with the quantity a ; then if a = 5 it follows that Q« = QJ. The reader will also remember that one of the most frequent operations in mathematical reasoning is to sub- stitute for a quantity its equal, as known either by assumed, natural, or self-evident conditions. Whenever a quantity appears twice over in a problem, we may apply what we learn of its relations in one place to its relations in the other. All reasoning in mathematics, as in other branches of science, thus involves the principle of treating equals equally, or similars similarly. In whatever way we employ quantitative reasoning in the remaining parts of this work, we never can desert the simple principle on which we first set out i^ r Reasoning by Inequalities. I have stated that all the processes of mathematical reasoning may be deduced from the principle of substi- tution. Exceptions to this assertion may seem to exist in the use of inequalities. The greater of a greater is undoubtedly a greater, and what is less than a less is certainly less. Snowdon is higher than the Wrekin, and Ben Nevis than Snowdon ; therefore Ben Nevis is higher than the Wrekin. But a little consideration discloses *»ufficient reason for believincr that even in such cases, 166 THE PRINCIPLES OF SCIENCE. [chap. Till.] PRINCIPLES OF NUMBER 167 ■ 8 i-i () where equality does not apparently enter, the force of the reasoning entirely depends upon underlying and implied equalities. In the first place, two statements of mere difference do not give any ground of inference. We Jeam nothing concerning the comparative heights of St. Paul's and Westminster Abhey from the assertions that they both differ in height from St. Peter's at Kome. We need some- thing more than inequality ; we require one identity in addition, namely the identity in direction of the two differences. Thus we cannot employ inequalities in the simple way in which we do equalities, and, when we try to express what other conditions are requisite, we find ourselves lapsing into the use of equalities or identities. In the second place, every argument by inequalities may be represented in the form of equalities. We express that a is greater than h by the equation a = 2>+jp, (l) where p is an intrinsically positive quantity, denoting the difference of a and b. Similarly we express that b is greater than c by the equation 6 = c + g, (2) and substituting for 6 in (i) its value in (2) we have a==c + q+p. (3) Now as p and q are both positive, it follows that a is greater than c, and we have the exact amount of excess specified. It will be easily seen that the reasoning con- cerning that which is less than a less will result in an equation of the form c z= a -- r " 8. Every argument by inequalities may then be thrown into the form of an equality ; but the converse is not true. We cannot possibly prove that two quantities are equal by merely asserting that they are both greater or both le^s than another quantity. From e >f and ^ >/, or e <f and g </,vfe can infer no relation between e and g. And if the reader take the equations a? = y = 3 and attempt to prove that therefore a; = 3, by throwing them into in- equalities, he will find it impossible to do so. From these considerations I gather that reasoning in arithmetic or algebra by so-called inequalities, is only an imperfectly expressed reasoning by equalities, and when we want to exhibit exactly and clearly the conditions of reasoning, we are obliged to use equalities explicitly. Just as in pure logic a negative proposition, as expressing mere difference, cannot be the means of inference, so inequalitv can never really be the true ground of inference. I do not deny that affirmation and negation, agreement and difference, equality and inequality, are pairs of equally fundamental relations, but I assert that inference is pos- sible only where affirmation, agreement, or equality, some species of identity in fact, is present, explicitly or implicitly. Arithmetical Reasoning, It may seem somewhat inconsistent that I assert number to arise out of difference or discrimination, and yet hold that no reasoning can be grounded on difference. Number of course, opens a most wide sphere for inference, and a bttle consideration shows that this is due to the unlimited senes of identities which spring up out of numerical abstraction. If six people are sitting on six chairs, there is no resemblance between the chairs and the people in logical character. But if we overlook all the qualities both of a chair and a person and merely remember that there are marks by which each of six chairs may be discriminated from the othei-s, and similarly with the people, then there arises a resemblance between the chairs and the people, and this resemblance in number may be the ground of inference. If on another occasion the chairs are hi ed by people again, we may infer that these people resemble the othera in number though they need not resemble them in any other points. Groups of units are what we really treat in arithmetic. The number Jive is really i + i + i + i + i, but for the sake of conciseness we substitute the more compact sign 5, or the name Jive. These names being arbitrarily im- posed m any one manner, an infinite variety of relations spring up between them which are not in the least arbitrary. If we define fonr as I + i + i + i, and >e as 1 + I + 1 + I + I, then of course it follows that five =>wr + I ; but it would be equally possible to take this latter equality as a definition, in which case one of tne former equalities would become an inference It is im THE PRINCIPLES OF SCIENCE. [OBAr. to :|'^ hardly requisite to decide how we define the names of numbers, provided we remember that out of the infinitely numerous relations of one number to others, some one relation expressed in an equality must be a definition of the number in question and the other relations imme- diately become necessary inferences. In the science of number the variety of classes which can be formed is altogether infinite, and statements of perfect generality may be made subject only to difficulty or exception at the lower end of the scale. Every existing number for instance belongs to the class m + 2; that is, every number must be the sum of another number and seven, except of coui-se the first six or seven numbers, negative quantities not being here taken into account. Every number is the half of some other, and so on. The subject of generalization, as exhibited in mathematical truths, is an infinitely wide one. In number we are only at the first step of an extensive series of generalizations. As number is general compared with the particular things numbered, so we have general symbols for numbers, and general symbols for relations between undetermined numbei*s. There is an unlimited hierarchy of successive generalizations. Numerically Definite Reasoning, It was first discovered by De Morgan that many argu- ments are valid which combine logical and numerical reasoning, although they cannot ^ included in the ancient logical formulas. He developed the doctrine of the " Numerically Definite Syllogism,** fully explained in his Formal Logic (pp. 141 — 170). Boole also devoted considerable attention to the detennination of what he called "Statistical Conditions," meaning the numerical conditions of logical classes. In a paper published among the Memoirs of the Manchester Literary and Pliilosophical Society, Third Series, voL IV. p. 330 (Session 1869—70), I have p* inted out that we can apply arithmetical calcula- tion to the Logical Alphabet. Having given certain logical conditions and the numbers of objects in certain classes, we can either determine the numbers of objects in other classes governed by those conditions, or can show what ▼iij.l PRINCIPLES OF NUMBER. 169 further dat^a are required to determine them. As an example of the kind of questions treated in numerical logic, and the mode of treatment, I give the following problem suggested by De Morgan, with my mode of representing its solution. '* For every man in the house there is a person who is aged ; some of the men are not aged. It follows that some persons in the house are not men."^ Now let A = person in house, B = male, C = aged. By enclosing a logical symbol in brackets, let us denote the number of objects belonging to the class indicated by the syn^bol. Thus let (A) = number of persons in house, (AB) = number of male persons in house, (ABC) ■= number of aged male persons in house, and so on. Now if we use w and w' to denote unknown numbers, the conditions of the problem may be thus stated according to my interpretation of the words — tliat IS to say, the number of persons in the house who are aged is at least equal to, and may exceed, the number of male persons in the house ; (ABc) = < (2) that is to say, the number of male persons in the house who are not aged is some unknown positive quantity. If we develop the terms in (i) by the Law of Duality (pp. 74, 81, 89), we obtain (ABC) + (ABc) = (ABC) + (AJC) - w, Subtractmg the common term (ABC) from each side and substituting for (ABc) its value as given in (2), we get at once (A5C) = w-^w\ and adding {Abe) to each side, we have (A6) = {Abe) -{■w + w\ Ihe meanmg of this result is that the number of persons in the house who are not men is at least equal io w + w\ and exceeds it by the number of persons in the house who are neither men nor aged (AJc). * Sylladui of a Proposed SysUm of Logic, p. 29. 170 THE PRINCIPLES OF SCIENCE. ToHAr. Till.] PRINCIPLES OP NUMBER. 171 J ril It should be understood that this solution applies only to the terms of the example quoted above, and not to the general problem for wliich De Morgan intended it to serve as an illustration. As a second instance, let us take the following ques- tion : — The whole number of voters in a borough is a ; the number against whom objections have been lodged by liberals is h; and the number against whom objections have been lodged by conservatives is c; required the number, if any, who have been objected to on both sides. Taking A = voter, B = objected to by liberals, C = objected to by conservatives, then we require the value of (ABC). Now the following equation is identically true — (ABC) = (AB) + (AC) + {Ahc) - (A). (i) For if we develop all the terms on the second side we obtain ^ABC) = (ABC) + (ABc) + (ABC) + (A6C) + (Abe) - (ABC) - (ABc) - (A6C) - {Khc) ; and striking out the corresponding positive and negative terras, we have left only (ABC) = (ABC). Since then (i) is necessarily tme, we have only to insert the known values, and we have (ABC) =.h-\-c-a-\- (A6c). Hence the number wlio have received objections from both sides is equal to the excess, if any, of the whole number of objections over the number of voters together with the number of voters who have received no objection {Xhc). The following problem illustrates the expression for the common part of any three classes: — The number of paupers who are blind males, is equal to the excess, if any, of the sum of the whole number of blind persons, added to the whole number of male persons, added to the number of those who being paupers are neither blind nor males, above the sum of the whole number of paupers added to the number of those who, not being paupers, are blind, and to the number of those who, not being paupers, are male. The reader is requested to prove the truth of the above statement (i) by his own unaided common sense; (2) by the Aristotelian Logic ; (3) by the method of numerical logic just expounded ; and then to decide which method is most satisfactory. Numeiical meaning of Logical Gonditicms. In many cases classes of objects may exist under spe- cial logical conditions, and we must consider how these conditions can be interpreted numerically. Every logical proposition gives rise to a corresponding numerical equation. Sameness of qualities occasions sameness of numbers. Hence if A = B denotes the identity of the qualities of A and B, we may conclude that (A) = (B). It is evident that exactly those objects, and those objects only, which are comprehended under A must be compre- hended under B. It follows that wherever we can draw an equation of qualities, we can di-aw a similar equation of numbers. Thus, from A = B = C we infer A-C; and similarly from (A) = (B) = (C), meanmg that the numbers of A's and C s are equal to the number of B's, we can infer . , (A) = (C). But, cunously enough, this does not apply to negative propositions and inequalities. For if A = B - D means that A is identical with B, which differs from D, it does not follow that (A) = (B) ^ (D). Two classes of objects may differ in qualities, and yet they niay agree in number. This point strongly confirms me in the opinion which I have already expressed, that all inference really depends upon equations, not differences. The Logical Alphabet thus enables us to make a com- plete analysis of any numerical problem, and though the symbolical statement may sometimes seem prolix, I con- j^ 172 THE PRINCIPLES OF SCIENCE. [chap. viii. ceive that it really represents the course which the mind must follow in solving the question. Although thought may outstrip the rapidity with which the symbols can be written down, yet the mind docs not really follow a different course from that indicated by the symbols. For a fuller explanation of this natural system of Numerically Definite Eeasoning, with more abundant illustrations and an analysis of De Morgan's Numerically Definite Syllogism, I must refer the reader to the paper^ in the Memoirs of the Manchester Literary and Philosophical Society, already mentioned, portions of which, however, have been embodied in the present section. The reader may be referred, also, to Boole's writin^^s upon the subject in the Laws of Thought, chap. xix. p. 295, and in a paper on "Propositions Numerically Definite," communicated by De Morgan, in 1868, to the Cambridge Philosophical Society, and printed in their Transactions;' vol. xi. part ii. » Jt has b«jn pointed out to me by Mr. C. J. Monroe, that section 14 (P- 339) of this paper is erroneous, and oaj?ht to be cancelled. The problem concerning the number of paupers illustrates the answer which should have been obtained. Mr. A. J. Ellis, F.R.S., ha<l previously observed that my solution in the paper of De Morgan's problem about " men in the house " did not answer the conditions mtended by De Morgan, and I therefore give in the text a more satisfactory solution. CHAPTER^ IX. i TH15 VAUIETY OP NATURE, OR THE DOCTRINE OP COMBINATIONS AND PERMUTATIONS. Nature may be said to be evolved from the monotony of non-existence by the creation of diversity. It is plau- sibly asserted that we are conscious only so far as we experience difference. Life is change, and perfectly uni- form existence would be no better than non-existence. Certain it is that life demands incessant novelty, and that nature, though it probably never fails to obey the same fixed laws, yet presents to us an apparently unlimited series of varied combinations of events. It is the work of science to observe and record the kinds and comparative numbers of such combinations of phenomena, occurring spontaneously or produced by our interference. Patient and skilful examination of the records may then disclose the laws imposed on matter at its creation, and enable us more or less successfully to predict, or even to regulate, the future occurrence of any particular combination. The Laws of Thought are the first and most important of all the laws which govern the combinations of pheno- mena, and, though they be binding on the mind, they may aLso be regarded as verified in the external world. The Logical Alphabet develops the utmost variety of things and events which may occur, and it is evident that as each new quality is introduced, the number of combi- nations is doubled. Thus four qualities may occur in 16 combinations; five qualities in 32; six qualities in 64; and 80 on. In general language, if n be the number of qualities, 2" is the number of varieties of things which \. 174 THE PRINCIPLES OF 80IEN0B. [OHAP. i may be fonned from them, if there be no conditions but those of logic. This number, it need hardly be said, increases after the first few terms, in an extraordinary manner, so that it would require 302 figures to express the number of combinations in which I,CXX) qualities might conceivably present themselves. If all the combinations allowed by the Laws of Thought occuned indifferently in nature, then science would begin and end with those laws. To observe nature would give us no additional knowledge, because no two qualities would in the long run be oftener associated than any other two. We could never predict events with more certainty than we now predict the throws of dice, and experience would be without usa But the universe, as actually created, presents a far different and much more interesting problem. The most superficial observation shows that some things are constantly associated with other things. The more mature our examination, the more we become convinced that each event depends upon the prior occurrence of some other series of events. Action and reaction are gradually discovered to underlie the whole scene, and an independent or casual occurrence does not exist except in appearance. Even dice as they fall are surely determined in their course by prior con- ditions and fixed laws. Thus the combinations of events which can really occur are found to be comparatively restricted, and it is the work of science to detect these restricting conditions. In the English alphabet, for instance, we have twenty- six letters. Were the combinations of such letters per- fectly free, so that any letter coidd be indifferently sounded with any other, the number of words which could be formed without any repetition would be 2^ — i, or 67,108,863, equal in number to the combinations of the twenty-seventh column of the Logical Alphabet, excluding one for the case in which all the letters would be absent. But the formation of our vocal organs prevents us from using the far greater part of these conjunctions of letters. At least one vowel must b*» present in each word ; more than two consonants cannot usually be brought together ; and to produce words capable of smooth utterance a number of other rules must be IX.] COMBINATIONS AND PERMUTATIONS. 17ft observed. To determine exactly how many words might exist in the English language under these circumstances, would be an exceedingly complex problem, the solution of which has never been attempted. The number of existing English words may perhaps be said not to exceed one hundred thousand, and it is only by investigating the com- binations presented in the dictionary, that we can learn the Laws of Euphony or calculate the possible number of words. In this example we have an epitome of the work and method of science. The combinations of natural phenomena are limited by a great number of conditions which are in no way brought to our knowledge except so far as they are disclosed in the examination of nature. It is often a very difficult matter to determine the num- bers of permutations or combinations which may exist under various restrictions. Many learned men puzzled themselves in former centuiies over what were called Protean verses, or verses admitting many variations in accordaace with tlie Laws of Metre. The most celebrated of these verses was that invented by Bernard Bauhusius, as follows : ^ — ** Tot tibi aiwt dotes, Virgo, quot sidera cceIo.* One author, Erioius Puteanus, filled forty-eight pages of a work in reckoning up its possible transpositions, making them only 1022. Other calculators gave 2196, 3276, 2580 as their results. Wallis assigned 3096, but without much confidence in the accuracy of his result.^ It required the skill of James Bernoulli to decide that the number of transpositions was 3312, under the condition that the sense and metre of the verse shall be perfectly preserved. In approacliing the consideration of the great Inductive problem, it is very necessary that we sliould acquire correct notions as to the comparative numbers of combinations which may exist under different circumstances. The doctrine of combinations is that part of mathematical science which applies numerical calculation to determine the numbers of combinations under various conditions. It is a part of the science which really lies at the base not only of other sciences, but of other branches of mathe- J Montucla, Ritioirty &c., vol. iii. p. 388. • Wallis, Of CombinaiioMf &c., p. iiQt 176 THE PRINCIPLES OF 8CIENCK [CHAB /] matics. The forms of algebraical expressions are deter- mined by the principles of combination, and Hindenburg recognised this fact in his Combinatorial Analysis. The greatest mathematicians have, during the last three cen- turies, given their best powers to the treatment of this subject ; it waa the favourite study of Pascal ; it early attracted the attention of Leibnitz, who wrote his curious essay, Be Arte Cmnbinatoria, at twenty years of age ; James Bernoulli, one of the very profoundest mathematicians, devoted no smaU part of his life to the investigation of the subject, as connected with that of Probability ; and in his celebrated work, Be Arte Gonjectandi, he has so finely described the importance of the doctrine of combinations, that I need offer no excuse for quoting his remarks at full length. " It is easy to perceive that the prodigious variety which appears both in the works of nature and in the actions of men, and which constitutes the greatest part of the beauty of the universe, is owing to the multitude of different ways in which its several parts are mixed with, or placed near, each other. But, because the number of causes that concur m producing a given event, or effect, is oftentimes so im- mensely great, and the causes themselves are so different one from another, that it is extremely difficult to reckon up all the different ways in which they may be arranged or combined together, it often happens that men, even of the best understandings and greatest circumspection, are guilty of that fault in reasoning which the writers on logic call tAe insufficient or imperfect enumeratian of parts or cases : insomuch that I will venture to assert, that this is the chief, and almost the only, source of the vast number of erroneous opinions, and those too very often in matters of great importance, which we are apt to form on all the subjects we reflect upon, whether they relate to the know- ledge of nature, or the merite and motives of human actions. It must therefore be acknowledged, that that art which affords a cure to this weakness, or defect, of our under- standiiigs, and teaches us so to enumerate all the possible ways in which a given number of things may be mixed and combined together, that we may be certain that we have not omitted any one arrangement of them that can IX.} COMBINATIONS AND PERMUTATIONS. 177 lead to the object of our inquiry, deserves to be considered as most emmently useful and worthy of our highest esteem and attention. And this is the business of the art or doctrine of combinations. Nor is this art or doctrine to be considered merely as a branch of the mathematical sciences. *or It has a relation to almost every species of useful know* ledge that the mmd of man can be employed upon It proceeds mdeed upon mathematical principles, in calculat- ing the number of the combinations of the things proposed • but by the conclusions that are obtained by it, the sagacity of the natural philosopher, the exactness of the historian, the skiU and judgment of the physician, and the prudence and foresight of the poUtician may be assisted; because the business of all these important professions is but to form reasonable conjectures concerning the several objects which engage their attention, and all wise conjectures are the results of a just and careful examination of the several different effects that may possibly arise from the causes tnat are capable of producing them." * Distinction of Combinaiims and Permutations. We must first consider the deep difference which exists between Combinations and Permutations, a difference in- volving important logical principles, and influencing the form of mathematical expressions. In permiUation we re- ?Z'%r"'T''' ^^ order, treating AB as a different group irom liA. In combination we take notice only of the presence or absence of a certain thing, and pay no regard to Its place m order of time or space. Thus tie 1^ letters a, e, m, n can form but one combination, but thev We have hitherto been dealing with purely logical oiip«- tions, involving only combinattjn of qualUief I W fully pointed out in more than one placl that! thonV our symbols could not b.it be written in order of plkce aKad m orier of time, the relations expressed had^rregarf to pla^or tune (pp 33, r 14). TTie Law of Commutativeness in fact, expresses the condition that in logic we deal with ITS THE PRINCIPLKS OP SCIENCK [cnAf. combinations, and the same law is true of all the processes of algebra. In some cases, order may be a matter of indifference ; it makes no difference, for instance, whether gunpowder is a mixture of sulphur, carbon, and nitre, or carbon, nitre, and sulphur, or nitre, sulphur, and carbon, provided that the substances are present in proper propor- tions and well mixed. But this indifference of order does not usually extend to the events of physical science or the operations of art. The change of mechanical energy into heat is not exactly the same as the change from heat into mechanical energy ; thunder does not indifferently precede and follow lightning ; it is a matter of some importance that we load, cap, present, and fire a rifle in this precise order. Time is the condition of all our thoughts, space of all our actions, and therefore both in art and science we are to a great extent concerned with permutations. Language, for instance, treats different permutations of letters as having different meanings. Permutations of things are far more numerous than combinations of those things, for the obvious reason that each distinct thing is regarded differently according to its place. Tlius the letters A, B, C, will make different permutations according as A stands first, second, or third ; having decided the place of A, there are two places between which we may choose for B ; and then there remains but one place for C. Accordingly the permuta- tions of these letters will be altogether 3x2x1 or 6 in number. With four things or letters. A, B, C, D, wo shall have four choices of place for the first letter, three for the second, two for the third, and one for the fourth, 80 that there will be altogether, 4x3x2x1, or 24 permutations. The same simple rule applies in all cases ; beginning with the whole number of things we multiply at each step by a number decreased by a unit. In general language, if n be the number of things in a combination, the number of permutations is 71 (n — i) (n — 2) 4.3.2. I. If we were to re-arrange the names of the days of the week, the possible arrangements out of which we should have to choose the new order, would be no less than 7 . 6 . 5 . 4 . 3 . 2 . I, or 5040, or, excluding the existing order, 5039. S IX.] COMBINATIONS AND PERMUTATIONS. Itt The reader will see that the numbers which we reach in questions of permutation, increase in a more extraordinary manner even than in combination. Each new object or term doubles the number of combinations, but increases the permutations by a factor continually growing. Instead of 2X2X2X2X we have 2X3X4X5X .and the products of the latter expression immensely exceed those of the former. These products of increasing factors are frequently employed, as we shall see, in ques- tions both of permutation and combination. They are technically called factorials, that is to say, the product of all integer numbers, from unity up to any number n is the factonal of n, and is often indicated symbolically by \n I give below the factorials up to that of twelve :— 24 = I . 2 . 3 . 4 120= I . 2 5 720 = I . 2 6 5,040 = [7 40,320 = L8 362,880 = L9 3,628,800 = |ip 39,916,800 - |ii 479.001,600 = (12 The factorials up to [36 are given in Rees's ' Cyclopedia, art. Cipher and the logarithms of factorials up to I265 TJ^^.A ^'^'T^ ^J ^^^ ^^^ ^^ *^« ^ble of logarithms published under the superintendence of the Society for he Biffusion of Useful Knowledge (p. 215). To express the factorial I265 would require 529 places of figures Many wnters have from time to time remarked upon tlie extraordinary magnitude of the numbers with which we deal in this subject. Tacquet calculated ^ that the twenty.four letters of the alphabet may be arranged in more than 620 thousand trillions of ordera ; and Schott estimated that if a thousand millions of men were em- ployed for the same number of years in writing out these arrangements, and each man filled each day forty pages with forty arrangements in each, they would not Imve accomplished the task, as they would have written onl} 584 thousand trillions instead of 620 thousand trillions. J Arithmetica Theoria. Ed. Amsteid. 1704. p C17 • Rees's Cyclopadia, art Cipher. -^ ^ ^ f N 2 M f ■ 180 THE PRINCIPLES OF SCIENCE. [CBAP. IX.J COMBINATIONS AND PERMUTATIONS. 181 In some questions the number of permutations may be restricted and reduced by various conditions. Some things in a group may be undistinguishable from others, so that change of order will produce no difference. Thus if we were to permutate the letters of the name Ann, according to our previous rule, we should obtain 3x2x1, or 6 orders ; but half of these arrangements would be identical with the other half, because the interchange of the two ns has no effect. The really different orders will "5 2 1 therefore be ^ — '— or 3, namely Ann, Nan, Nna., In the word ntility there are two I's and two t'&, in respect of both of which pairs the numbers of permutations must be halved. Thus we obtain 7.6.5.4.3.2.1 or 1260, as 1 . 2 . 1 . 2 the number of permutations. The simple rule evidently is — when some things or letters are undistinguished, proceed in the first place to calculate all the possible permutations as if all were different, and then divide by th6 numbers of possible permutations of those series of things which are not distinguished, and of which the permutations have therefore been counted in excess. Thus since the word Utilitarianisvi contains fourteen letters, of which four are i*s, two as, and two fs, the number of distinct arrangements will be found by dividing the factorial of 14, by the factorials of 4, 2, and 2, the result being 908,107,200. From the letters of the word Mississippi we can get in like manner , 1= j- or 34,650 permutations, which is not the one- [4 XLi X [2 thousandth part of what we should obtain were all the letters different Calculation 0/ Number of Combinations. Although in many questions both of art and science we need to calculate the numbtir of permutations on account of their own interest, it far more frequently happens in scientific subjects that the)' possess but an indirect interest. As I have already pointed out, we almost always deal in the logical and mathematical sciences witli combinaiions, and varie^ of order enters only through the inherent imperfections of our symbols and modes of calculation. Signs must be used in some order, and we must withdraw our attention from this order before the signs correctly represent the relations of things which exist neither before nor after each other. Now, it often happens that we cannot choose all the combinations of things, without first choosing them subject to the accidental variety of order, and we must then divide by the number of possible variations of order, that we may get to the true number of pure combinations. Suppose that we wish to determine the number of ways in which we can select a group of three letters out of the alphabet, without allowing the same letter to be repeated. At the first choice we can take any one of 26 letters ; at the next step there remain 25 letters, any one of which may be joined with that already taken ; at the third step there will be 24 choices, so that apparently the whole number of ways of choosing is 26 x 25 x 24. But the fact that one choice succeeded another has caused us to obtain the same combinations of letters in different orders ; we should get, for instance, a, p, r at one time, and^, r, a at another, and every three distinct letters will appear six times over, because three things can be arranged in six permutations. To get the number of combinations, then, we must divide the whole number of ways of choosing, by six, the number of permutations of three things, 26 X 25 X 24 2x3 or 2,600. obtaining — ^ It is apparent that we need the doctrine of combina- tions in order that we may in many questions counteract the exaggerating effect of successive selection. If out of a senate of 30 persons we have to choose a committee of 5, we may choose any of 30 first, any of 29 next, and so on, in fact there will be 30 x 29 x 28 x 27 x 26 selections; but as the actual character of the members of the committee will not be affected by the accidental order of their selec- tion, we divide byi X2X3X4X5, and the possible number of different committees will be 142,506. Similarly if we want to calculate the number of ways in which the eight major planets may come into conjunction, it is evi- dent that they may meet either two at a time or three at a time, or four or more at a time, and as nothing is said iU 1*0 h 182 THE PRINCIPLES OF SCIENCE. [chap. «.l COMBINATIONS AND PERMUTATIONS. 18ft I f I the relative order or place in the conjunction, we require the number of combinations. Now a selection of 2 out of 8 is possible in ;^l or 28 ways ; of 3 out of 8 in ?^ ^•2 1.2.3 or 56 ways ; of 4 out of 8 in ^'^'^'^ or 70 ways ; and it may be similarly shown that for 5, 6, 7, and 8 planets, meeting at one time, the numbers of ways are 56, 28, 8, and I. Thus we have solved the whole question of the' variety of conjunctions of eight planets ; and adding all the numbers together, we find that 247 is the utmost possible number of modes of meeting. In general algebraic language, we may say that a group of m things may be chosen out of a total number of n things, in a number of combinations denoted by the formula n . (n- I) (n~2) (n-3) (n-m + i ) ''2 . 3.4.... n The extreme importance and significance of this formula seems to have been firet adequately recognised by Pascal, although its discovery is attributed by him to a friend, M.' de Gani^res.^ We shall find it perpetually recurring in questions l)oth of combinations and probability, and throughout the formulae of mathematical analysis traces of its influence may be noticed. Th€ AHthmeticcU Triangle, The Arithmetical Triangle is a name long since given to a series of remarkable numbers connected with the subject we are treating. According to Montucla « '* this triangle is in the theory of combinations and changes of order, almost what the table of Pythagoras is in ordinary arithmetic, that is to say, it places at once under the eyes the numbers re- quired in a multitude of cases of this theory." As early is 1544 Stifels had noticed the remarkable properties of these numbers and the mode of their evolution. Briggs, the inventor of the common system of logarithms, was *80 struck with their importance that he called them the > (Buvres CompUtes de Pascal (1865), vol. iii. p. 302. Montucla states the name «s De Gruiires, Histoire des Mathematiqua, vol iii P- 389- ' UUtoire des Mathefimtiquet, vol iii. p. 378. Abacus Panchrestus. Pascal, however, was the first who wrote a distinct treatise on these numbers, and gave them the name by which they are still known. But Pascal did not by any means exhaust the subject, and it remained for James Bernoulli to demonstrate fully the importance of the figurate numbers, as they are also called. In his treatise De Arte Conjectandi, he points out their applica- tion in the theoiy of combinations and probabilities, and remarks of the Arithmetical Triangle, " It not only con- tains the clue to the mysterious doctrine of combinations, but it is also the ground or foundation of most of the im- portant and abstruse discoveries that have been made in the other branches of the mathematics." ^ The numbers of the triangle can be calculated in a very easy manner by successive additions. We commence with unity at the apex ; in the next line we place a second unit to the right of this ; to obtain the third line of figures we move the previous line one place to the right, and add them to the same figures as they were before removal ; we can then repeat the same process ad infinitum. The fourth line of figures, for instance, contains i, 3, 3, i ; moving them one place and adding as directed we obtain : — Fourth line . . . I 3 3 I I 3 3 I Fifth line .... I 4 6 4 I I 4 6 4 I Sixth line .... I 5 10 10 5 I Carrying out this simple process through ten more stepa we obtain the first seventeen lines of the Arithmetical Triangle as printed on the next page. Theoretically speaking the Triangle must be regarded as infinite in extent, but the numbers increase so rapidly that it soon becomes impracticable to continue the table. The longest table of the numbers which I have found is in Fortia's ** IVait^ des Progressions " (p. 80), where they are given up to t)ie fortieth line and the ninth column. » Bernoulli, De Arte Ckmjeetandiy translated by Fi-ancis Maaerea. lioudon, I795i P* 75* ' I N( f ? 184 s ! THE PRINCIPLES OF SCIENCE [CHAF. o u H w If s I- c • IS 0) tSl '^ O to fO w roc I 5-2 O I- fl5 « N « O O is- ti xr\^ ^\o a»oo o *o r-s. 65 a S e a S l<s a a o .a U3 00 «o o row N li^O *• f) *^ »>. "^ n* "^ CO WOO*-vON*-0 M '*• OM^ O O vOMNOvor4Mfits.f( rooo M ton M-.vo oxoo o O'O •-• N fJ "<»■ t^ O tOOO •* ^ O O »nvO ^ O »0 O O •• W ro M^ao N vO N OC vd w^v_ ►* ■< M W ro '«• »n ^"^Jsg V *< ►< « W to «♦ u^n6 r^ Ov O umm op vp tA) tovp 00 M to O e ^ ►* W to ^ »0>0 tM» 0« O M (i to ^ u>0 •^Q|rj •" « to ^ mvp l>»QO Ov O •■ « to V w%>0 IX.] COMBINATIONS AND PERMUTATIONS. 185 Examining these numbers, we find that they are con- nected by an unlimited series of relations, a few of the more simple of which may be noticed. Each vertical column of numbers exactly corresponds with an oblique series descending from left to right, so that the triangle is perfectly symmetrical in its contents. The first column contains only units; the second column contains the natural numhers, I, 2, 3, &c. ; the third column contains a remarkable series of numbers, I, 3, 6, 10, 15, &c., which have long been called tJie triangular numbers, because they correspond with the numbers of balls which may be arranged in a triangular form, thus — o o o o o o 000 o o o 000 0000 o o 000 0000 00000 The fourth column contains the pyramidal numbers, so called because they correspond to the numbers of equal balls which can be piled in regular triangular pyramids. Their differences are the triangular numbers. The numbers of the fifth column have the pyramidal numbers for their differences, but as there is no regular figure of which they express the contents, they have been arbitrarily called the trianguli-triangular numbers. The succeeding columns have, in a similar manner, been said to contain the trianguli-pyramickU, the pyramidi-pyramidal numbers, and so on.^ From the mode of formation of the table, it follows that the differences of the numbers in each column will be found in the preceding column to the left. Hence the second differenceSy or the differences of differences^ will be in the second column to the left of any given column, the third differences in the third column, and so on. Thus we may say that unity which appears in the first column is the first difference of the numbers in the second column ; the second difference of those in the third column ; the third difftrence of those in the fourth, and so on. The triangle is seen to be a complete classification of all numbers according as they have unity for any of their differences. Since each line is formed by adding the previous line ( Willis's Algtbra, Discourse of Combinations, ^q., p. 109. \^' I 186 THE PRINCIPLES OF SCIENGR [coAi*. till to itself, it is evident that the sum of the numbers in each horizontal line must be double the sum of the numbers in the line next above. Hence we know, without making the additions, that the successive sums must be i, 2, 4, 8, 16, 32, 64, &c., the same as the numbers of combinations in the Logical Alphabet. Speaking generally, the sum of the numbers in the nth line will be 2*-\ Again, if the whole of the numbers down to any line be added together, we shall obtain a number less by unity than some power of 2; thus, the first line gives i or 2^— I ; the first two lines give 3 or 2^— 1 ; the finst three lines 7 or 2^ — i ; the first six lines give 63 or 2* — i ; or, speaking in general language, the sum of the first n lines is 2* — I. It follows that the sum of the numbers in any one line is equal to the sum of those in all the preceding lines increased by a unit. For the sum of the nth line is, as ah-eady shown, 2'^\ and the sum of the first ti— i lines is 2*^' — I, or less by a unit. This account of the properties of the figurate numbers does not approach completeness ; a considerable, probably an unlimited, number of less simple and obvious relations might be traced out. Pascal, after giving many of the properties, exclaims ^ : "Mais j'en laisse bien plus que je n'en donne ; c'est une chose Strange combien il est fertile en propri^tes! Chacun pent s'y exercer." The arith- metical triangle may be considered a natural classification of numbers, exhibiting, in the most complete manner, their evolution and relations in a certain point of view. It is obvious that in an unlimited extension of the triangle, each number, with the single exception of the number two, has at least two places. Though the properties above explained are highly curious, the greatest value of the triangle arises from the fact that it contains a complete statement of the values of the formula (p. 182), for the numbers of combinations of m things out of n, for all possible values of ni and n. Out of seven things one may be chosen in seven ways, and seven occurs in the eighth line of the second column. The combinations of two things chosen out of seven are 7x6 or 21, which is the third number in the eighth I X 2 * UHuvr^ CompUks^ vol, iii. p, 251. ■'f I J n> «■ IX.] COMBINATIONS AND PERMUTATIONS. 187 line. The combinations of three things out of seven are 7x6x5 1X2X3 ^^ ^5' which appeai-s fourth in the eighth line. In a similar manner, in the fifth, sixth, seventh, and eighth columns of the eighth line I find it stated in how many ways I can select combinations of 4, 5, 6, and 7 things out of 7. Proceeding to the ninth line, I find in succession the number of ways in which I can select i, 2, 3, 4, 5, 6 7, and 8 things, out of 8 things. In general lan^age, if I wish to know in how many ways m things can be selected in combinations out of n things, I must look in the n + I**' line, and take the wi + i** number, as the answer. In how many ways, for instance, can a sub- committee of five be chosen out of a committee of nine. The answer is 126, and is the sixth number in the tenth line; it will be found equal to 9 ■ « » 7 . 6^5^ ^^^^y^ our formula (p. 182) gives. * -3.4.5 The full utility of the figurate numbers will be more apparent when we reach the subject of probabilities, but I may give an illustration or two in this place. In how many ways can we arrange four pennies as regards head and taU ? The question amounts to asking in how many ways we can select o, i, 2, 3, or 4 heads, out of 4 heads, and the fifth line of the triangle gives us the complete answer, thus — We can select No head and 4 tails in i way. „ I head and 3 tails in 4 ways. M 2 heads and 2 tails in 6 ways. >, 3 heads and i tail in 4 ways. „ 4 heads and o tail in i way. The total number of different cases is 16, or 2* and when we come to the next chapter, it will be found that these nuinbers give us the respective probabilities of all throws with four pennies. I ^ve in p. 181 a calculation of the number of ways in which eight planets can meet in conjunction ; the reader will find all the numbers detailed in the ninth line of the anthmetical triangle. The sum of the whole line is 2» or 256; but we must subtract a unit for the case where no planet appears, and 8 for the 8 cases in which only one planet appears; so that the total number of conjunctions \ 188 THE PRINCIPLES OF SCIENCE. [OOAP. is 2* — I — 8 or 247. If an organ has eleven stops we find in the twelfth line the numbers of ways in which we can draw them, i, 2, 3, or more at a time. Thus there are 462 ways of drawing five stops at once, and as many of drawing six stops. The total number of ways of varying the sound is 2048, including the single case in which no stop at all is drawn. One of the most important scientific uses of the arith- metical triangle consists in the information which it gives concerning the comparative frequency of diveigencies from an average. Suppose, for the sake of argument, that all persons were naturally of the equal stature of five feet, but enjoyed during youth seven independent chances of growing one inch in addition. Of these seven chances, one, two, three, or more, may happen favourably to any individual; but, as it does not matter what the chances are, so that the inch is gained, the question really turns upon the number of combinations of o, I, 2, 3, &c., things out of seven. Hence the eighth line of the triangle gives us a complete answer to the question, as follows : — Out of every 128 people — »# One person would have the stature of 7 persons 21 persons 35 persons 35 persons 21 persons 7 persons I person it t» u n M Feet 5 5 5 5 5 5 5 5 IdcKm O I 2 3 4 5 6 7 By taking a proper line of the triangle, an answer may be had under any more natural supposition. This theory of comparative frequency of divergence fi-om an average, was first adequately noticed by Quetelet, and has lately been employed in a very interesting and bold manner by Mr. Francis Galton,^ in his remarkable work on " Hereditary Genius." We shall afterwards find that the theory of error, to which is made the ultimate appeal in cases of quantitative investigation, is founded upon the * See also Galton's Lecture at the Royal Institution, 27th February, 1874 ; Catalogue of the Spocial Loan Collection of Scientific Instru- ments, South Kensington, Nob. 48 49 ; and Galton, FhiiosophictU Ma^iuine, January 1875, «.] COMBINATIONS AND PERMUTATIONS. 1B9 comparative numbers of combinations as displayed in the triangle. Connection between the Arithmetical TrtOngle and the Logical Alphabet, There exists a close connection between the arithmetical triangle described in the last section, and the series of combinations of letters called the Logical Alphabet. The one is to mathematical science what the other is to logical science. In fact the figurate numbers, or those exhibited in the triangle, are obtained by summing up the logical combinations. Accordingly, just as the total of the numbers in each line of the triangle is twice as great as that for the preceding line (p. 186), so each column of the Alphabet (p. 94) contains twice as many combinations as the preceding one. The like correspondence also exists between the sums of all the lines of figures down to any particular line, and of the combinations down to any particular column. By examining any column of the Logical Alphabet we find that the combinations naturally group themselves according to the figurate numbers. Take the combinations of the letters A, B, C, D ; they consist of all the ways in which I can choose four, three, two, one, or none of the four letters, filling up the vacant spaces with negative terms. There is one combination, ABCD, in which all the positive letters are present ; there are four combinations in each of which three positive letters are present; six in which two are present ; four in which only one is present ; and, finally, there is the single case, abed, in which all positive letters are absent. These numbers, i, 4, 6, 4, i, are those of the fifth line of the arithmetical triangle, and a like correspondence will be found to exist in each column of the Ix)gical Alphabet. Numerical abstraction, it has been asserted, consists in overlooking the kind of difference, and retaining only a consciousness of its existence (p. 158). While in logic, then, we have to deal with each combination as a separate kind of thing, in arithmetic we distinguish only the classes which depend upon more or less positive terms being \ IdO THE PRINCIPLES OP SCIENCE. ■ rif' < [cBAt. present, and the numbers of these classes immediately produce the numbers of the arithmetical triangle It may here be pointed out that there are two modes in which we can^alculate the whole number of combinations 01 certam thmgs. Either we may take the whole number at once as shown in the Ixjgical Alphabet, in which case the number will be some power of two, or else we may calculate successively, by aid of permutations, the number ot combinations of none, one, two, three things, and so on. Hence we arrive at a necessary identity between two have' °""'^'*^- '" ^^'"^ ^^e «^ ^^^^^ <'hings we shall 2 = I 4- 1 + 4_^ 3 , 4 > 3 > 2 4 3. 2 ^i I '•2^i.2.3"^"i.2.34- In a general form of expression we shall have ' 1.2 ^ 1.2.3 • ' the terms being continued until they cease to have any value. Thus we arrive at a proof of simple cases of the Binomial Theorem, of which each column of the Logical Alphabet IS an exemplification. It may be shown that aU other mathematical expansions likewise arise out of simple processes of combination, but the more complete considera- tion of this subject must be deferred to another work. Possible Variety of Nature and Art We cannot adequately understand the difficulties which beset us in certain branches of science, unless we have some clear idea of the vast numbers of combinations or permutations which may be possible under certain con- ditions Thus only can we learn how hopeless it would be to attempt to treat nature in detail, and exhaust the whole number of events which might arise. It is instruc- tive to consider, in the first place, how immensely great are the numbers of combinations with which we deal in many arts and amusements. In dealing a pack of cards, the number of hands, of thirteen cards each, which can be produced is evidently 52 X 51 X 50 X . . . X 40 divided by i x 2 x ^ x lit or 635,013,559,600. But in whist four hands are' simut' 11.] COMBINATIONS AND PERMUTATIONS. 191 taneously held, and the number of distinct deals becomes so vast that it would require twenty-eight figures to express it If the whole population of the world, say one thousand millions of persons, were to deal cards day and night, for a hundred million of years, they would not in that time have exhausted one hundred-thousandth part of the pos- sible deals. Even with the same hands of cards the play may be almost infinitely varied, so that the complete variety of games at whist which may exist is almost incalculably great. It is in the highest degree improbable that any one game of whist was ever exactly like another, except it were intentionally so. The end of novelty in art might well be dreaded, did we not find that nature at least has placed no attainable limit, and that the deficiency will lie in our inventive faculties. It would be a cheerless time indeed when all possible varieties of melody were exhausted, but it is readily shown that if a poal of twenty-four bells had been rung continuously from the so-called beginning of the world to the present day, no approach could have been made to the completion of the possible changes. Nay, had every single minute been prolonged to 10,000 years, still the task would have been unaccomplished.^ As regards ordinary melodies, the eight notes of a single octave give more than 40,000 permutations, and two octaves more than a million millions. If we were to take into account the semitones, it would become apparent that it is impossible to exhaust the variety of music. When the late Mr. J. S Mill, in a depressed state of mind, feared the approaching exhaustion of musical melodies, he had certainly not bestowed sufficient study on the subject of permutations. Similar considerations apply to the possible number of natural substances, though we cannot always give precise numerical results. It was recommended by Hatehett ' that a systematic examination of all alloys of metals should be carried out, proceeding from the binary ones to more complicated ternary or quaternary ones. He can hardly have been aware of the extent of his proposed 1 * Wallis, Of Combinations^ p. 116, quoting Vossius. * Philosophical Transactiont (1803). vol. xciii. p. 193. 19S THE PRINCIPLES OF SCIENC?E. [OBAP. IX.] COMBINATIONS AND PERMUTATIONS. H i inquiry. If we operate only upon thirty of the known metals, the number of binary alloys would be 435, of ternary alloys 4060, of quaternary 27,405, without paying regard to the varying proportions of the metals, and only regarding the kind of metal. If we varied all the ternary alloys by quantities not less than one per cent., the number of these alloys would be 11445,060. An ex- haustive investigation of the subject is therefore out of the question, and unless some laws connecting the proper- ties of the alloy and its components can be discovered, it is not apparent how our knowledge of them can ever be more than fragmentary. The possible variety of definite chemical compounds, again, is enormously great. Chemists have already ex- amined many thousands of inorganic substances, and a still greater number of organic compounds ; ^ they have nevertheless made no appreciable impression on the number which may exist. Taking the number of ele- ments at sixty-one, the number of compounds contain- ing different selections of four elements each would be more than half a million (521,855). As the same elements often combine in many different proportions, and some of them, especially carbon, have the power of forming an almost endless number of compounds, it would hardly be possible to assign any limit to the number of chemical compounds which may be formed. There are branches of physical science, therefore, of which it is unlikely that scientific men, with all their industry, can ever obtain a knowledge in any appreciable degree approaching to completeness. Higher Orders of Variety. The consideration of the facts already given in this chapter will not produce an adequate notion of the pos- sible variety of existence, unless we consider the com- parative numbers of combinations of different orders. By a combination of a higher order, I mean a combination of groups, which are themselves groups. The immense numbers of compounds of carbon, hydrogen, and oxygen, ^ Uohuann'a Introduction to ChemxHry^ p. 36. 193 described in organic chemistry, are combinations of a second order, for the atoms are groups of groups. The wave of sound produced by a musical instrument may be regarded as a combination of motions ; the body of sound proceeding from a large orchestra is therefore a complex aggregate of sounds, each in itself a complex combination of movements. All literature may be said to be developed out of the difference of white paper and black ink. From the unlimited number of marks which might be cliosen we select twenty-six conventional letters. The pronounceable combinations of letters are probably some trillions in number. Now, as a sentence is a selection of words, the possible sentences must be inconceivably more numerous than the words of which it may be composed. A book is a combination of sentences, and a library is a combination of books. A library, therefore, may be regarded as a com- bination of the fifth order, and the powers of numerical expression would be severely tasked in attempting to express the number of distinct libraries which might be constructed. The calculation, of course, would not be possible, because the union of letters in words, of words in sentences, and of sentences in books, is governed by conditions so complex as to defy analysis. I wish only to point out that the infinite variety of literature, existing or possible, is all developed out of one fundamental differ- ence. Galileo remarked that all truth is contained in the compass of the alphabet. He ought to have said that it is all contained in the difference of ink and paper. One consequence of successive combination is that the simplest marks will suffice to express any information. Francis Bacon proposed for secret writing a biliteral cipher, which resolves all letters of the alphabet into permutations of the two letttirs a and b. Thus A was aaaaa, B acuiab, X habab, and so on.i In a similar way, as Bacon clearly saw, any one difference can be made the ground of a code of signals ; we can express, aa he says, omnia per omnia. The Morse alphabet uses only a Bucceasion of long and short marks, and other systems of telegraphic language employ right and left strokes A single lamp obscured at various intervals, long or » Works, ediiod by Shaw, vol i pp. 141—145, quoted in Bees' EneyclopcBdia, art Cipher. II u / 194 THE PRINCIPLES OF SCIENCE. [chap. short, may be made to spell out any words, and with two lamps, distinguished by colour, position, or any other circumstance, we could at once represent Bacon's biliteral alphabet. Babbage ingeniously suggested that every lighthouse in the world should be made to spell out its own name or number perpetually, by flashes or obscurations of various duration and succession. A system like that of Babbage is now being applied to lighthouses in the United Kingdom by Sir W. Thomson and Dr. John Hopkinson. Let us calculate the numbers of combinations of dif- ferent orders which may arise out of the presence or absence of a single mark, say A. In these figures | A|A | lAI I I !AJ l_J I we have four distinct varieties. Form them into a group of a higher order, and consider in how many ways we may vary that group by omitting one or more of the component parts. Now, as there are four parts, and any one may be present or absent, the possible varieties will be2X2X2X2, or i6in number. Form these into a new whole, and proceed again to create variety by omitting any one or more of the sixteen. The number of pos- sible changes will now be 2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2, or 2", and we can repeat the process again and again. We are imj^ining the creation of objects, whose numbers are represented by the successive orders of the powers of two. At the fii-st step we have 2 ; at the next 2*, or 4 ; at the third 2^ , or 16, numbers of very moderate amount. 2 2 Let the reader calculate the next term, 2* , and he will be surprised to find it leap up to 65,536. But at the next step he has to calculate the value of 65,536 tu'o*3 multiplied together, and it is so great that we could not possibly compute it, the mere expression of the result requiring 19,729 places of figures. But go one step more and we pass the bounds of all reason. The sixth order of the powers of ttuo becomes so great, that we could not even express the number of figures required in writing it down, without using about 19,729 figures for the purpose. The successive orders of the powers of two have then the II >■ IX.] COMBINATIONS AND PERMUTATIONS. 195 following values so far as we can succeed in describing them : — First order .... 2 Second order . . . . 4 Third order .... 16 Fourth order .... 65,536 Fifth order, number expressed by 19,729 figures. Sixth order, number expressed by figures, to express the number of which figures would require about .... 19,729 figures. It may give us some notion of infinity to remember that at this sixth step, having long surpassed all bounds of intuitive conception, we make no approach to a limit. Nay, were we to make a hundred such steps, we should be as far away as ever from actual infinity. It is well worth observing that our powers of expression rapidly overcome the possible multitude of finite objects which may exist in any assignable space. Archimedes showed long ago, in one of the most remarkable writings of antiquity, the Liber de Arencc Numero, that the grains of sand in the world could be numbered, or rather, that if numbered, the result could readily be expressed in arith- metical notation. Let us extend his problem, and ascertain whether we could express the number of atoms which could exist in the visible universe. The most distant stars which can now be seen by telescopes— those of the sixteenth magnitude— are supposed to have a distance of about 33*900,000,000,000,000 miles. Sir W. Thomson has shown reasons for supposing that there do not exist more than from 3 x lo^* to lo^* molecules in a cubic centimetre of a solid or liquid substance.* Assuming these data to be true, for the sake of argument, a simple calculation enables us to show that the abnost inconceivably vast sphere of our stellar system if entirely filled with solid matter, would not contain more than about 6S x 10^ atoms, that is to say, a number requiring for its expression 92 places of figures. Now, this number would be im- mensely less than the fifth order of the powers of two. In the variety of logical relations, which may exist » Nature, vo'. i P. 553 o 2 il !1 196 THE PRINCIPLES OF SCIENCE. [qbat. n, between a certain number of logical terms, we also meet a case of higher combinations. We have seen (p. 142) that with only six terms the number of possible selections of combinations is 18446,744,073,709,551,616. Considering that it is the most common thing in the world to use an argument involving six objects or terms, it may excite some surprise that the complete investigation of the relations in which six such terms may stand to each other, shoidd involve an almost inconceivable number of cases. Yet these numbers of possible logical relations belonc[ only to the second order of combinations. ;;• ^ r i CHAPTER X. THE THEORY OP PROBABILITY. The subject upon which we now enter must not be reg:arded as an isolated and curious branch of speculation. It is the necessary basis of the judgments we make in the prosecution of science, or the decisions we come to in the conduct of ordinary affairs. As Butler truly said, " Pro- bability is the very guide of life." Had the science of numbers been studied for no other purpose, it must have been developed for the calculation of probabilities. All our inferences concerning the future are merely probable, and a due appreciation of the degree of probability depends upon a comprehension of the principles of the subject. I am convinced that it is impossible to expound the methods of induction in a sound manner, without resting them upon the theory of probability. Perfect knowledge alone can give certainty, and in nature perfect knowledge would be infinite knowledge, which is clearly beyond our capacities. We have, therefore, to content ourselves with partial knowledge — knowledge mingled with ignorance, producing doubt A great difficulty in this subject consists in acquiring a precise notion of the matter treated. What is it that we number, and measure, and calculate in the theory of pro- babilities ? Is it belief, or opinion, or doubt, or knowledge, or chance, or necessity, or want of art ? Does probability exist in the things which are probable, or in the mind which regards them as such ? The etymology of the name lends us no assistance : for, curiously enough, probable is ultimately the same word as provable, a good instance of one word becoming differentiated to two opposite meanings. \\ h f \ II I ; ,1 i ! I .t '<W IM THE PRINCIPLES OF SCIENCE. [ORAP. Chance cannot be the subject of the theory, because there is really no such thing as chance, regarded as pro- ducing and governing events. The word cliance signifies falling, and the notion of falling is continually used as a simile to express uncertainty, because we can seldom pre- dict how a die, a coin, or a leaf will fall, or when a bullet will hit the mark. But everyone sees, after a little reflection, that it is in our knowledge the deficiency lies, not in the certainty of nature's laws. There is no doubt in lightning as to the point it shall strike; in the greatest storm there is nothing capricious ; not a grain of sand lies upon the beach, but infinite knowledge would account for its lying there ; and the course of every falling leaf is guided by the principles of mechanics which rule the motions of the heavenly bodies. Chance then exists not in nature, and cannot coexist with knowledge; it is merely an expression, as Laplace remarked, for our ignorance of the causes in action, and our consequent inability to predict the result, or to bring it about infallibly. In nature the happening of an event has been pre-determinod from the first fashioning of the universe. Prohahility belongs wholly to the mind. This is proved by the fact that different minds may regard the very same event at the same time with widely different degrees of probability. A steam-vessel, for instance, is missing and some persons believe that she has sunk in mid -ocean; others think differently. In the event itself there can be no such uncertainty ; the steam-vessel either has sunk or has not sunk, and no subsequent discussion of the probable nature of the event can alter the fact. Yet the probability of the event will really vary from day to day, and from mind to mind, according as the slightest information is gained regarding the vessels met at sea, the weather prevailing there, the signs of wreck picked up, or the previous condition of the vessel. Probability thus belongs to our mental condition, to the light in which we regard events, the occurrence or non-occurrence of which is certain in themselves. Many writers accordingly have asserted that probability is concerned with degree or quantity of belief. De Morgan says,^ " By degree of proba- * Formal Lo^^ p. 172. «.] THE THEORY OF PROBABILITY. 199 .'I bility we really mean or ought to mean degree of belief." The late Professor Don kin expressed the meaning of probability as " quantity of belief ; " but I have never felt satisfied with such definitions of probability. The nature of belief is not more clear to my mind than the notion which it is used to define. But an all-sufficient objection is, that the theory does not measure what the belief is, but what it ought to be. Few minds think in close accordance with the theory, and there are many cases of evidence in which the belief existing is habitually different from what it ought to be. Even if the state of belief in any mind could be measured and expressed in figures, the results would be worthless. The value of the theory consists in coiTecting and guiding our belief, and rendering our states of mind and consequent actions harmonious with our knowledge of exterior conditions. This objection has been clearly perceived by some of those who still used quantity of belief as a definition of probability. Thus De Morgan adds — "Belief is but another name for imperfect knowledge." Donkin has well said that the quantity of belief is " always relative to a particular state of knowledge or ignorance; but it must be observed that it is absolute in the sense of not being relative to any individual mind; since, the same information being presupposed, all minds ought to dis- tribute their belief in the same way." ^ Boole seemed to entertain a like view, when he described the theory as engaged with "the equal distribution of ignorance;"* but we may just as well say that it is engaged with the equal distribution of knowledge. I prefer to dispense altogether with this obscure word belief, and to say that the theory of probability deals with qtiantiiy of knowledge, an expression of which a precise explanation and measure can presently be given. An event is only probable when our knowledge of it is diluted with ignorance, and exact calculation is needed to discriminate how much we do and do not know. The theory has been described by some writers as professing to evolve knowledge out of ignorance ; but as Donkin admirably remarked, it is really " a method of avoiding the erection ' Philosophical Afagazine, 4th Series, vol. i. p. 355. * Transa4;tumt of the Royal Society of Edinburghy vol. x»i. paxt 4, li ' i 900 THE PRINCIPLES OF SCfENOE. II If) II J [chap. hv ^^}f 'i.P"" ^gnorance." It defines rational expectation by measuring he comparative amounts of knowledge and Jgno^nce and teaches us to regulate our Jionf w*tS regard to future events in a way which wUl. in the lone run. lead to the least disappointment It is, a.s llpl^ happUy said, ffood stnseredn^ to calculation. This theorv appears to me the noblest creation of intellect andH passes ^y conception how two such men as Auguste Oomte and J. S. Mill could be found depreciating it and vainlv questioning its validity. To eulojise the theory ou^h to be as needless as to eulogise reason itself. Fundamental Principles of the Thevry. The calculation of probabilities is really founded as I conceive upon the principle of reasoning set forth in pre we kLwTr'- ^' "•"'' "-^"^ «*1'"J^ equally and7h^ we know of one case may be affirmed of every case reserablmg ,t m the necessaiy circumstances. The theo^ consists in putting similar ca-ses on a par, and distribS equally among them whatever knowled^ wf ^sS Throw a penny into the air, and consider^whlt wHuow with regard to its way of falling. We know that it 3 certainly fall upon a side, so that either head or tail w be uppermost; but as to whether it will be head or Ta our knowledge IS equally divided. Whatever we know concerning head, we know also concerning tail, so that we ThlwV'-T" ^"^ ^xP^cting one more "than the othlr The least predominance of belief to either side would bo irrational; it would consist in treating unequal^ thinS of which our knowledge is equal. ^ ^ The theory does not require, as some writers have ei-roneously supposed, that we should first ^Zi^Z sXr^'c^''? ^-l'"^ ^'^^'y of *»>« events ^Z\Z sidcnng. So far as we can examine and measure the oTpXbihr'Th'eT'^ "^ "'""^^ ""' of ZTph'ere 01 proDabUity. The theory comes into play where ionor ance begins, and the knowledge we possJL requfres tote distributed over many cases. Nor docs the thTr^ show that the com will fall as often on the one side TZ otheT Tt IS almost impossible that this should hap^n because »ome inequality in the form of the coin, or S u^form ■ «J THE THEORY OP PROBABILITY. 201 manner in throwing it up, is almost sure to occasion a slight preponderance in one direction. But as we do not previously know in which way a preponderance will exist we have no reason for expecting head more than tail Our state of knowledge wiU be changed should we throw up the com many times and register the results. Every throw gives us some slight information as to the probable tendency of the coin, and in subsequent calculations we must take this into account. In other cases experience might show that we had been entirely mistaken ; we mi^ht expect that a die would faU as often on each of the lix sides as on each other side in the long run ; trial might show that the die was a loaded one, and falls most often on a particular face. The theory would not have misled us • it treated correctly the information we had, which is all that any theory can do. It may be asked, as Mill asks, Why spend so much trouble m calculating from imperfect data, when a little trouble would enable us to render a conclusion certain bv actual trial ? Why calculate the probabUity of a measure- ment being correct, when we can try whether it is correct ? But I shall fully point out in later parts of this work that m measurement we never can attain perfect coincidence Two measurements of the same base line in a survey may show a difference of some inches, and there may be no means of knowing wliich is the better result. A third measurement would probably agree with neither. To select any one of the measurements, would imply that we knew it to be the most nearly coiTect one, which we do not In this state of ignorance, the only guide is the theory of probabihty, which proves that in the lon<T run the mean of divergent results wUl come most neariy to the truth. In aU other scientific operations whatsoever perfect knowledge is impossible, and when we have ex- hausted all our instrumental means in the attainment of truth, there is a margin of error which can only be safely treated by the principles of probability. The method which we employ in the theory consists in calculating the number of all the cases or events concerning, which our knowledge is equal. If we have the slightest reason for suspecting that one event is more likely to occur than another, we should take this knowledge into k i( \l iJl r ., i f02 THE PRINCIPLES OF SCIENCE. [chap. C accouni This being done, we must determine the whole number of events which are, so far as we know, equally likely. Thus, if we have no reason for supposing that a penny will fall more often one way than another, there are two cases, head and tail, equally likely. But if from trial or otherwise we know, or think we know, that of loo throws 55 will give tail, then the probability is measured by the ratio of 55 to loa The mathematical formulae of the theory are exactly the same as those of the theory of combinations. In this latter theory we determine in how many ways events may be joined together, and we now proceed to use this know- ledge m calculating the number of ways in which a certain event may come about. It is the comparative numbers of ways m which events can happen which measure their comparative probabQities. If we throw three pennies into the air, what is the probability that two of them will fall taa uppermost ? This amounts to askin.^ in how many possible ways can we select two tails out%f three compared with the whole number of ways in which the coins can be placed. Now, the fourth line of the Arith- metical Triangle (p. 184) gives us the answer. The whole number of ways in which we can select or leave three thinrrs IS eight, and the possible combinations of two things at^'a time IS three ; hence the probability of two tails^'is the ratio of three to eight. From the numbers in the trian<Tle we may similarly draw all the following probabilities :-^ One combination gives o tail. Probability J. Three combinations gives i tail Probability f . Three combinations give 2 tails. Probability f . One combination gives 3 tails. Probability J. We can apply the same considerations to the imaginary causes of the difference of stature, the combinations of which were shown in p. 188. There are altogether 128 ways m which seven causes can be present or absent. Now, twenty-one of these combinations give an addition of two inches, so tliat the probability of a person under the circumstances being five feet two inches is ^. The probability of five feet three inches is yVff ; of five feet one inch ^ ; of five feet ^4^, and so on. Thus the eighth line of the Arithmetical Triangle gives all the probabilities arising out of the combinations of seven causes. *.l THE THEORY OF PROBABILITY. 3Rld Bules for tlie CalctclcUion of Probabilities, I will now explain as simply as possible the rules for calculating probabilities. The principal rule is as follows : — Calculate the number of events which may happen independently of each other, and which, as far as is known, are equally probable. Make this number the denominator of a fraction, and take for the numerator the number of such events as imply or constitute tne happening of the event, whose probability is required. Thus, if the letters of the word Roma be thrown down casually in a row, what is the probability that they will form a significant Latin word ? The possible arrange- ments of four letters are 4 X 3 x 2 x i, or 24 in number (p. 178), and if all the arrangements be examined, seven of these will be found to have meaning, namely Roma, ramo, oram, mora, maro, arm^, and amor. Hence the probability of a significant result is ^, We must distinguish comparative from absolute pro- babilities. In drawing a card casually from a pack, there is no reason to expect any one card more than any other. Now, there are four kings and four queens in a pack, so that there are just as many ways of drawing one as the other, and the probabilities are equal. But there are thirteen diamonds, so that the probability of a king is to that of a diamond as four to thirteen. Thus the probabili- ties of each are proportional to their respective numbers of ways of happening. Again, I can draw a king in four ways, and not draw one in forty-eight, so that the pro- babilities are in this proportion, or, as is commonly said, the odiis against drawing a king aro forty-eight to four. The odds are seven to seventeen in favour, or seventeen to seven against the letters R,o,m,a, accidentally forming a significant word. The odds are five to three against two tails appearing in three throws of a penny. Conversely, when the odds of an event are given, and the probability is required, take the odds in favour of the event for numerator, and the sum of the odds for denominator. It is obvious that an event is certain when all the com- binations of causes which can take place produce that event If we represent the probability of such event ' i h 904 THE PRINCIPLES OP SCIENCE. [crap. according to our rule, it gives the ratio of some number to itself, or unity. An event is certain not to happen when no possible combination of causes gives the event, and the ratio by the same rule becomes tliat of o to some' number Hence it follows that in the theory of probability certainty IS expressed by i, and impossibility by o ; but no mystical meaning should be attached to these symbols, as they merely express the fact that all or tw possible combinations give the event. By a compound event, we mean an event which may be decomposed into two or more simpler events. Thus the firing of a gun may be decomposed into puUin^ the trigger, the fall of the hammer, the explosion of the cap, &C. In this example the simple events are not independcTU, because if the trigger is pulled, the other events will under proper conditions necessarily follow, and their probabilities are therefore the same as that of the first event Events are independent when the happenin<T of one does not render the other either more or le^ probable than before. Thus the death of a person is neither more nor less probable because the planet Mars happens to be visible. When the component evente are independent, a simple rule can be given for calculatincr the probabihty of the compound eveut, thuH^Multiply together the fractions expressing the probabilities of the independent component events. The probability of throwing tail twice with a penny is * X i, or i ; the probabUity of throwing it three times running is i x J x J, or J ; a result agreeing with that obtained m an apparently diiierent manner (p. 202). In fact, when we multiply together the denominators, we get the whole number of ways of happening of the com- pound event, and when we multiply the numerators, we get the number of ways favourable to the required event Probabilities may be added to or subtracted from each other under the important condition that the events in question are exclusive of each other, so that not more than one of them can happen. It might be argued that, since the probability of throwing head at the first trial is i, and at the second tnal also i, the probability of throwing it in the firat two throws is ^ + J, or certainty. Not only is this result evidentlv absurd, but a repetition of the process x.] THE THEORY OF PROBABILITY. 805 would lead us to a probability of i^ or of any greater number, results which could liave no meaning whatever. The probability we wish to calculate is that of one head in two throws, but in our addition we have included the case in which two heads appear. The tme result is J + J x ^ or }, or the probability of head at the first throw, added to the exclusive probability that if it does not come at the first, it will come at the second. The greatest difficulties of the theory arise from the confusion of exclusive and unexclusive alternatives. I may remind the reader that the possibility of unexclusive alternatives was a point previously discussed (p. 68), and to the reasons then given for considering alternation as logically unexclusive, may be added the existence of these difficulties in the theory of probability. The erroneous result explained above really arose from overlooking the fact that the expression " head first throw or head second throw " might include the case of head at both throws. The Logical Alphabet in questions of Probability. When the probabilities of certain simple events are given, and it is required to deduce the probabilities of compound events, the Logical Alphabet may give assist- ance, provided that there are no special logical conditions so that all the combinations are possible. Thus, if there be three events, A, B, C, of which the probabilities are, a, ^, 7, then the negatives of those events, expressing the absence of the events, will have the probabilities i — a, i —fi, 1—7. We have only to insert these values for the letters of the combinations and multiply, and we obtain the probability of each combination. Thus the probability of ABC is aJ3y; of Abe, a(l - /9)(l - 7). We can now clearly distinguish between the probabilities of exclusive and unexclusive events. Thus, if A and B are events which may happen together like rain and high tide, or an earthquake and a stoi-m, the probability of A or B happening is not the sum of their separate probabilities. For by the Laws of Thought we develop A -I- B into AB'|'A5«|»aB, and substituting a and 0, the probabili- ties of A and B respectively, we obtain a.l3-\-a.(l — /9)-h (l— a).)9 or a+fi—a.fi. But if events are incompossible 206 THE PRINCIPLES OF SCIENCE. [chap. ht/ or incapable of happening together, like a clear sky and rain, or a new moon and a full moon, then the events are not really A or B, but A not-B, or B not- A, or in symbols Ab •!• aB. Now if we take /a = probability of A6 and V = probability of aB, then we may add simply, and the probability of Ab I- aB is /* + v. Let the reader carefully observe that if the combi- nation AB cannot exist, the probability of Ab is not the product of the probabilities of A and b. When certain combinations are logically impossible, it is no lonj^er allowable to substitute the probability of each term for the term, because the multiplication of probabilities pre- supposes the independence of the events. A large part of Boole's Laws of Thought is devoted to an attempt to overcome this difficulty and to produce a Oeneral Method in Probabilities by which from certain logical conditions and certain given probabilities it would he possible to deduce the probability of any other combinations of events under those conditions. Boole pursued his task with wonderful ingenuity and power, but after spending much study on his work, I am compelled to adopt the conclusion that his method is fundamentally erroneous. As pointed out by Mr. Wilbraham.^ Boole obtained his results by an arbitrary assumption, which is only the most probable, and not the only possible assumption. Tiie answer obtained is therefore not the real probability, which is usually indeterminate, but only, as it were, the '■ most probable probability. Certain problems solved by. Boole are free from logical conditions and therefore may admit of valid answers. These, as I have shown,* may be solved by the combinations of the Logical Alphabet, but the rest of the problems do not admit of a determinate answer, at least by Boole's method. Comparison of the Huory with Experience. The Laws of Probability rest upon the fundamental prin- ciples of reasoning, and cannot be really negatived by any ' Phihtophical Magazine, 4th Scries, vol. vii. p. 465 ; vol. viii p. 01. Alemoirs of the Manchester Literary and Philosophical Society, 3rd Senei, vol. iv. p. 347 «•) THE THEORY OP PEOBABILITT. 207 fS i^^^Tu""^ ^' '"^''' '•^PPe" tl>«^ a person should always throw a coin head up^rmost, and Appear Lottti "i f K ^° "^^ ^y «'^'^»'=^ The theory w^d not be falsified, because it contemplates the possibiUty of might be counter to all that is probable; the whole course of events might seem to be'^m complete contra! diction to what we should expect, and yet a casual con^ illsT Z •1,'^f?'! ""S*"* ^ '^' ^'^ explanation iTk ittrih^« i fi "J," '*""*, ^^^"^ coincidences, which we Sn.1 /^ ^^"^ °^ "**"'^' """^ <J"« to the accidental conjunction of phenomena in the cases to which our attention is directed. All that we can learn from finite experience is capable, according to the theory of probabili- hS ° ^•^^^'"g ««' ""d it is only infinite experience that could assure us of any inductive truths r,.;^'Ji ?""?, t™«' the probability that any extreme runs of luck will occur is so excessively slight that it would be absurd seriously to expect their^occSn^nca I il^f "P^'f'^i*''/"' "Stance, that any whist player S f7i! P'*^"* '" *°y *^» g»°>«« ^here the distri^ bution of the cards was exactly the same, by pure accident ^^'J^P'e V* *^"?8 ^ * 1*'^° ai'-ays losing at a game of pure chance, is wholly unknown. Coincidences li^l ru w.°°' i"'P'"«iWe. as I have said, but they T «•>, ""''^^IJ^ .that the lifetime of any person, or indeed mbSv Tr^'^'^'y' ^"^ """^^ive any 'ap^rSble probability of their being encountered. Whenever we make any extensive series of trials of chance results, as in throwing a die or coin, the probability is great th^t the 7^^ "v '^'*' "''^'y ^'* ^^^ prodictiofs yielded by «wZ' I'^'^'' agreement must not be expected, for that, as the theory shows, is highly improbSle. Several a^.^f fK*"" ^«° ""«le to test, in this way, the accorf- W^ffhf"""! f*^ experience. Buflfon caused the first l?^J; """*^-^y " ^"'"•S "^^^ ^ho thr^w a coin many times in succession, and he obtained 1992 tails to 2048 iwn ttui P^P'* "' De Moi^n repeated the trial for Us own satisfaction, and obtained 2044 tails to 2048 heads. In both cases the coincidence with theory is as cltse as could 208 THE PRINCIPLES OF SCIENCE. [chap. 4 Quetelet also tested the theory in a rather more com- plete manner, by placing 20 black and 20 white balls in an urn and drawing a ball out time after time in an indifferent manner, each ball being replaced before a new drawing was made. He found, as might be expected, that the greater the number of drawings made, the more nearly were the white and black balls equal in number. At the ter- mination of the experiment he had registered 2066 white and 2030 black balls, the ratio being 102.1 I have made a series of experiments in a third manner, which seemed to me even more interesting, and capable of more extensive trial Taking a handful of ten coins, usually shillings, I threw them up time after time, and registered the numbers of heads which appeared each time. Now the probability of obtaining 10, 9, 8, 7, &c., heads is proportional to the number of combinations of 10, 9, 8, 7, &c., things out of 10 things. Consequently the results ought to approximate "to the numbers in the eleventh line of the Arithmetical Triangle. I made altogether 2C48 throws, in two sets of 1024 throws each, and the numbers obtained are given in the following table : — Theoretical Nuuibera. First Series. Second Series. Average. Divergence. xo Heads o Tail 9 M I .« 8 >• • *. 7 M 3 *> 6 M 4 .. 5 t» 5 >• 4 •• " »t 3 ». 7 » » - « .. > >* 9 M M 10 M t 10 45 uo •to •5« •10 xao 4S 10 X 3 la 57 xtx "57 aoi IXI S« •I X •3 73 183 190 832 X97 ti9 S» «5 X i "5 il + I + 7* + so + 6 -'U — XI + 8 Ttttelt xaa4 ias4 xoa4 XOM — I He whole number of single throws of coins amounted to 10 X 2048, or 20480 in all, one half of which or 10^340 should theoretically give head. The total number » Letters on the Theory of FrobabUitits. translated by Downes, 1840. PP- 30, 37. *.] THE THEORY OF PROBABILITY. 209 of heads obtained was actually 10,353, or 5222 in the first series, and 5 131 in the second. The coincidence with theory is pretty close, but considering the large number of throws there is some reason to suspect a tendency in favour of heads. The special interest of this trial consists in the ex- hibition, in a practical form, of the results of Bernoulli's theorem, and the law of error or divergence from the mean to be afterwards more fully considered. It illus- trates the connection between combinations and permu- tations, which is exhibited in the Arithmetical Triangle, and which underlies many important theorems of scienca' Frcbahle Deductive Arguments. With the aid of the theory of probabilities, we may extend tlie sphere of deductive argument. Hitherto we have treated propositions as certain, and on the hypo- thesis of certainty have deduced conclusions equally ^^\^^^' ^^^ *'^ie information on which we reason in ordinary life is seldom or never certain, and almost all reasoning is reaUy a question of probability. We oufrht therefore to be fully aware of the mode and degree^'in which deductive reasoning is affected by the theory of probability, and many persons may be surprised at the results which must be admitted. Some controversial writers appear to consider, as De Morgan remarked ^ that an inference from several equally probable premises is itself as probable as any of them, but the true result is very different. If an argument involves many proposi- tiODs, and each of them is uncertain, the conclusion will be of very little force. The validity of a conclusion may be regarded as a com- pound event, depending upon the premises happening to be true ; thus, to obtain the probability of the conclusion we must multiply together the fractions expressing the probabilities of the premises. If the probability is k that A is B, and also J that B is C, the conclusion that A is C on the ground of these premises, is J x ^ or J. Similarly if there be any number of premises requisite to the establish- ' Eneyclopadta Metropolitana, art. ProbabiliUesy i>. 396. i u U1 lio TOT PRINCIPLES OP SCIENCE. [cBAt. «.] THE THEORY OP PROBABILITY. 811 > ment of a conclusion and their probabilities be p, q, r, &c., the probability of the conclusion on the ground of these premises isp x q x r x This product kos but a small value, unless each of the quantities p, q, &c, be nearly unity. But it is particularly to be noticed that the probability thus calculated is not the whole probability of the con- clusion, but that only which it derives from the premises in question. Whately's ^ remarks on this subject might mislead the reader into supposing that the calculation is coiijpleted by multiplying together the probabilities of the premises. But it has been fully explained by De Morgan * that we must take into account the antecedent probability of the conclusion ; A may be C for other reasons besides its being B, and as he remarks, " It is difficult, if not impossible, to produce a chain of argument of which the rcasoner can rest the result on those arguments only." The failure of one argument does not, except under special circumstances, disprove the truth of the conclusion it is intended to uphold, otherwise there ai-e few truths which could survive the ill-considered arguments adduced in their favour. As a rope does not necessarily break because one or two strands in i fail, so a conclusion may depend upon an endless number of considerations besides those imme- diately in view. Even ^hen we have no other informa- tion we must not consider a statement as devoid of all probability. The true expression of complete doubt is a ratio of equality between the chances in favour of and against it, and this ratio is expressed in the probability J. Now if A and C are wholly unknown things, we have no reason to believe that A is C rather than A is not C. The antecedent probability is then i. If we also have the probabilities that A is B, J and that B is C, J we have no right to suppose that the probability of A being C is re- duced by the argument in its favour. If the conclusion is true on its own grounds, the failure of the argument docs not affect it ; thus its total probability is its antecedent probability, added to the probability that this failing, tlie new argument in question establishes it There is a pro- » Elements of LogtCj Book III. sections 1 1 and 18. • Encyelopadia Mdropoliiana^ art ProhahiliiiUy p. 40a r bability J that we shall not require the special argument; a probability ^ that we shall, and a probability J that the argument does in that case establish it. Thus the com- plete result is J + ^ X i, or |. In general language, if a be the probability founded on a particular ai'gument, and c the antecedent probability of the event, the general result is I - (i - a) (i - c), or a + c - oc. We may put it still more generally in this way : — Let a, 6, c, &c. be the probabilities of a conclusion grounded on various arguments. It is only when all the arguments fail that our conclusion proves finally untrue ; the proba- bilities of each failing are respectively, i — a, i — 6, i ~ c, &C. ; the probability that they will all fail is (i - a)(i _ h) (l - c)... ; therefore the probability that the conclusion will not fail is i - (i - a){i - h){i - c)... &c. It follows that every argument in favour of a conclusion, however flimsy and slight, adds probability to it When it is unknown whether an overdue vessel has foundered or not, every slight indication of a lost vessel will add some proba- bility to the belief of its loss, and the disproof of any particular evidence will not disprove the event We must apply these principles of evidence with great care, and observe that in a great proportion of cases the adducing of a weak argument does tend to the disproof of its conclusion. The assertion may have in itself great inherent improbability as being opposed to other evidence or to the supposed law of nature, and every reasoner may be assumed to be dealing plainly, and putting forward the whole force of evidence which he possesses in its favour. If he brings but one argument, and its probability a is small, then in the formula i - (i- a)(i - e) both a and c are small, and the whole expression has but little value. The whole effect of an ai^ument thus turns upon the question whether other arguments remain, so that we can introduce other factors (1-6), (i -rf), &c., into the above expression. In a court of justice, in a publication having an express purpose, and in many other cases, it is doubtless right to assume that the whole evidence considered to have any value as regards the conclusion asserted, is put forward. To assign the antecedent probability of any proposition, may be a matter of difficulty or impossibility, and one P 2 aaai^.;:. A^ 212 THE PTIINCIPLKS OF SCIKNOK [cRir. »•] THE THEORY OF PROBABILITY. tu wiUi which logic and the theory of probability have little concern. From the general body of science in our posses- sion, we must in each case make the best judgment we can. But in the absence of all knowledge the probability should be considered = J, for if we make it less than this we incline to believe it false rather than true. Thus, before we possessed any means of estimating the magnitudes of tlie fixed stars, the statement that Sirius was greater than the sun had a probability of exactly ^ ; it was as likely that it would be greater as that it would be smaller ; and so of any other star. This was the assumption which Michell made in his admirable speculations.^ It might seem, indeed, that as every proposition expresses an agreement, and the agreements or resemblances between phenomena are infinitely fewer than the differences (p. 44), every pro- position should in the absence of other information be infinitely improbable. But in our logical system every term may be indifferently positive or negative, so that we express under the same form as many differences as agree- ments. It is impossible therefore that we shoidd have any reason to disbelieve rather than to believe a statement about things of which we know nothing. We can hardly indeed invent a proposition concerning the truth of which we are absolutely ignorant, except when we are entirely ignorant of the terms used. If I ask the reader to assign the odds that a " Platythliptic Coefficient is positive " he will hardly see his way to doing so, unless he regard them as even. The assumption that complete doubt is properly ex- pressed by ^ lias been called in question by Bishop Terrot,* who proposes instead the indefinite symbol J; and he considers that "the d priori probability derived from absolute ignorance has no effect upon the force of a subsequently admitted probability." But if we grant that the probability may have any value between o and i, and that every separate value is equally likely, then n and I — » are equally likely, and the average is always J. Or we may take j) , dp to express the probability that our ' Philosophical Transactions (1767). Abridg. vol. xil. p. 435. ^ Tra»$actuyiu 0/ tks Edinburgh Fhilosojpkieal Soeieli^, voL xxi P-375- I ^^ estimate concerning any proposition should lie be ween p and p + dp. The complete pi-obability of the .proposition is then the integral taken between the limits i Jind o, or again J. Diffkidties of the Theory. The theory of probability, though undoubtedly true, requires very careful application. Not only is it a branch of mathematics in which oversights are frequently com- mitted, but it is a matter of great difficulty in many cases, to be sure that the formula correctly represents the data of the problem. These difficulties often arise from the logical complexity of the conditions, which might be, perhaps, to some extent cleared up by constantly bearing in mind the system of combinations as developed in the Indirect Ix>gical Method. In the study of probabilities, mathematicians had unconsciously employed logical pro- cesses far in advance of those in possession of logicians, and the Indirect Method is but the full statement of these processes. It is very curious how often the most acute and power- ful intellects have gone astray in the calculation of probabilities. Seldom was Pascal mistaken, yet he in- augurated the science with a mistaken solution.^ Leibnitz fell into the extraordinary blunder of thinking that the number twelve was as probable a result in the throwing of two dice as the number eleven.^ In not a few cases the false solution first obtained seems more plausible to the present day than the correct one since demonstrated. James Bernoulli candidly records two false solutions of a problem which he at first thought self-evident ; and he adds a warning i against the risk of error, especiidly when we attempt to reason on this subject without a rigid adherence to methodical rules and symbols. Montmort was not free from similar mistakes. D'Alembert con- stantly fell into blundei-s, and could not perceive, for instance, that the probabilities would be the same when * Montucla, FUstoire des MathJmatiqueSf \o\. iii. p. 386. ■ Leibuitz Opera^ Dutens* Edition, vol. vi part i. p. 217. Tod- hunter's History of tlte Theory of Probability, p. 48. To the latter work I am indebted for many of the statements in the text. \\ 214 THE PRINCIPLES OF SCIENCE. [CBAP. W coins are thrown successively as when thrown simul- taneously. Some men of great reputation, such as Ancillon, Moses Mendelssohn, Garve, Auguste Comte,* Poinsot, and J. S. MiU,'^ have so far misapprehended the theoiy, as to question its value or even to dispute its validity. The erroneous statements ahout the theory given in the earlier editions of Mill's System of Logic were par- tially withdrawn in the later editions. Many persons have a fallacious tendency to believe that when a chance event has happened several times together in an unusual conjunction, it is less likely to happen again. D'Alembert seriously held that if head was thrown three times running with a coin, tail would more probably appear at the next trial.' Bequelin adopted the same opinion, and yet there is no reason for it whatever. If the event be really casual, what has gone before cannot in the slightest degree influence it As a matter of fact, the more often a casual event takes place the more likely it is to happen again; because there is some slight empirical evidence of a tendency. The source of the fallacy is to be found entirely in the feelings of surprise with whicli we witness an event happening by chance, in a manner which seems to proceed from design. Misapprehension may also arise from overlooking the difference between permutations and combinations. To throw ten heads in succession with a coin is no more unlikely than to throw any other particular succession of heads and tails, but it is much less likely than five heads and five tails without regard to their order, be- cause there are no less than 252 different particular throws which will give this result, when we abstract the difference of order. Difliculties arise in the application of ,the theory from our habitual disregard of slight probabilities. We are obliged practically to accept truths as certain which are nearly so, because it ceases to be worth while to calculate the difference. No punishment could be inflicted if absolutely certain evidence of guilt were required, and as * Positive Philosophy f translated by Martineau, vol. ii. p. 12a ■ SvsUm of Loqioy bk. iii. chap. 18, 5th Ed. vol. ii. p. 61. ■ Montucla, Histaire, vol iii p. 405 ; Todhunter, p. 263. '] THE THEORY OF PROBABILITY. 215 Locke remarks, " He that will not stir till he infallibly knows the business he goes about will succeed, will have but little else to do but to sit still and perish."* There is not a moment of our lives when we do not lie under a slight danger of death, or some most terrible fate. There is not a single action of eating, drinking, sitting down, or standing up, which has not proved fatal to some person. Sevei-al philosophers have tried to assign the Umit of the probabilities which we regard as zero ; Buffon named tv.^tttt' because it is the probability, practically disregarded, that a man of 56 years of age will die the next day. Pascal remarked that a man would be esteemed a fool for hesitating to accept death when three dice gave sixes twenty times running, if his reward in case of a different result was to be a crown ; but as the chance of death in question is only i -i- 6^, or unity divided by a number of 47 places of figures, we may be said to incur greater risks every day for less motives. There is far greater risk of death, for instance, in a game of cricket or a visit to the rink. Nothing is more requisite than to distinguish carefully between the truth of a theory and the truthful application of the theory to actual circumstances. As a general rule, events in nature and art will present a complexity of relations exceeding our powers of treatment The intricate action of the mind often intervenes and renders complete analysis hopeless. If, for instance, the probability that a marksman shall hit the target in a single shot be i in 10, we might seem to have no difficulty in calculating the probability of any sucession of hits ; thus the proba- bility of three successive hits would be one in a thousand. But, in reality, the confidence and experience derived from the first successful shot would render a second success more probable. The events are not really independent, and there would generally be a far greater preponderance of runs of apparent luck, than a simple calculation of probabilities could account for. In some persons, however, a remarkable series of successes will produce a degree of excitement rendering continued success almost impossible. Attempts to apply the theory of probability to the I 1 Essay concerning Uwman Unier standing y bk. iv. clu 14. § n. £16 THE PRINCIPLES OF SCIENCE. [chap. results of judicial proceedings have proved of little value, siraply because the conditions are far too intricate. As Laplace said, " Tant de passions, d'int^rets divers et de circonstances compliquent les questions relatives d ces objets, qu'elles sont presque toujours insolubles." Men acting on a jury, or giving evidence before a court, are subject to so many complex influences that no mathema- tical formulas can be framed to express the real conditions. Jurymen or even judges on the bencli cannot be regarded as acting independently, with a definite probability in favour of each delivering a correct judgment. Each man of the jury is more or less influenced by the opinion of the others, and there are subtle effects of character and manner and strength of mind which defy analysis. Even in physical science we can in comparatively few cases apply the theory in a definite manner, because the data required are too complicated and difficult to obtain. But such failures in no way diininish the truth and beauty of the theory itself ; in reality there is no branch of science in which our symbols can cope with the complexity of Nature. As Donkin said, — " I do not see on what ground it can be doubted that every definite state of belief concerning a proposed hypo- thesis, is in itself capable of being represented by a nume- rical expression, however difficult or impracticable it may be to ascertain its actual value. It would be very difficult to estimate in numbera the vis viva of all the particles of a human body at any instant ; but no one doubts that it is capable of numerical expression." ^ The difficulty, in short, is merely relative to our know- ledge and skill, and is not absolute or inherent in the subject We must distinguish between what is theo- retically conceivable and what is practicable with our present mental resources. Provided that our aspirations are pointed in a right direction, we must not allow them to be damped by the consideration that they pass beyond what can now be turned to immediate usa In spite of its immense difficulties of appliciition, and the aspersions which have been mistakenly cast upon it, the theory of probabilities, I repeat, is the noblest, as it will in course » Philosophical Magazine, 4th Series, vol. i. p. 354 «.] THE THEORY OF PROBABILITY. 217 of time prove, perhaps the most fruitful branch of mathe- matical science. It is the very guide of life, and hardly can we take a step or make a decision of any kind without correctly or incorrectly making an estimation of proba- bilities. In the next chapter we proceed to consider how the whole cogency of inductive reasoning rests upon pro- babilities. The truth or untruth of a natural law, when carefully investigated, resolves itself into a high or low degree of probability, and this is the case whether or not we are capable of producing precise numerical data. )| ii if-i li'i CHAPTER XL PHILOSOPHY OF INDUCTIVE INFEHENCE. We have inquired into the nature of perfect induction, whereby we pass backwards from certain observed com- binations of events, to the logical conditions governing such combinations. We have also investigated the grounds ot that theory of probability, whicli must be our guide when we leave certainty behind, and dilute knowledge with Ignorance. There is now before us the difficult task of endeavouring to decide how, by the aid of that theory, we can ascend from the facts to the laws of nature ; and may then with more or less success anticipate the future couree of events. All our knowledge of natural objects must be ultimately derived from observation, and the diflicult question arises— How can we ever know anything which we have not directly observed through one of our senses, the apertures of the mind ? The utility of reason- ing is to assure ourselves that, at a determinate time and place or under specified conditions, a certain phenomenon wiU be observed. When we can use our senses and per- ceive that the phenomenon does occur, reasoning is super- fluous If the senses cannot be used, because the event 18 in the future, or out of reach, how can reasoning take their place ? Apparently, at least, we must infer the un- known from the known, and the mind must itself create an addition to the sum of knowledge. But I hold that it is quite impossible to make any real additions to the con- tents of our knowledge, except through new impressions upon the senses, or upon some aeai, pf feeling. I shall OH. XI.] PHITX)SOPHY OF INDUCTI^T: INFERENCK 219 attempt to show that inference, whether inductive or deductive, is never more than an unfolding of the contents of our exx)erience, and that it always proceeds upon the assumption that the future and the unperceived will be governed by the same conditions as the past and the perceived, an assumption which will often prove to be mistaken. In inductive as in deductive reasoning the conclusion never passes beyond the premises. Keasoning adds no more to the implicit contents of our knowledge, than the arrangement of the specimens in a museum adds to the number of those specimens. Arrangement adds to our knowledge in a certain sense : it allows us to perceive the similarities and peculiarities of the specimens, and on the assumption that the museum is an adequate representation of nature, it enables us to judge of the prevailing forms of natural objects. Bacon's first aphorism holds perfectly true, that man knows nothing but what he has observed, provided that we include his whole sources of experience, and the whole implicit contents of his knowledge. In- ference but unfolds the hidden meaning of our observations, and the theory of probability shows how far we go beyond our data in assuming that new specimens will resemble the old ones, or that the future may be regarded as proceeding unifoiinly with the past. Varums Gla,sses of Inductive TtuOls. It will be desirable, in the first place, to distinguish between the several kinds of truths which we endeavour to establish by induction. Although there is a certain common and universal element in all our processes of reasoning, yet diversity aiises in their application. Similarity of condition between the events from which we argue, and those to which we argue, must always be the ground of inference; but this similarity may have regawl either to time or place, or the simple logical combination of events, or to any conceivable junction of circumstances involving quality, time, and place. Haying met with many pieces of substance possessing ductility and a bright yellow colour, and having discovered, by perfect induction, that they all possess a hifh si>ecific 320 THE PRINCIPLES OF SCIENCE. [cHir. S\ gravity, and a freedom from the corrosive action of acids, we are led to expect that every piece of substance, possess- ing like ductility and a similar yellow colour, will have an equally high specific gravity, and a like freedom from corrosion by acids. This is a case of the coexistence of qualities ; for the character of the specimens examined alters not with time nor place. In a second class of cases, time will enter as a prin- cipal ground of similarity. When we hear a clock pendulum beat time after time, at equal intervals, and with a uniform sound, we confidently expect that the stroke will continue to be repeated uniformly. A comet having appeared several times at nearly equal intervals, we infe*r that it will probably appear again at the end of another like interval. A man who has returned home evening after evening for many years, and found his house stand*^ mg, may, on like grounds, expect that it Avill be standing the next evening, and on many succeeding evenings. Even the continuous existence of an object in an unaltered state, or the finding again of that which we have hidden, is but a matter of inference depending on experience. A still larger and more complex class of cases involves the relations of space, in addition to those of time and quality. Having observed that every triangle drawn upon the diameter of a circle, with ite apex upon the circum- ference, apparently contains a right angle, we may ascertain that all triangles in similar circumstances will contain right angles. This is a case of pure space reason- ing, apart from circumstances of time or quality, and it seems to be governed by different principles of reasoning. I shall endeavour to show, however, that geometrical reasoning differs but in degree from that which applies to other natural relations. The Relation of Cause and Effect. In a very large part of the sci^tific investigations which must be considered, we deal with events which follow from previous events, or with existences which succeed existences. Science, indeed, might arise even were material nature a fixed and changeless whole. Endow mind with the power to travel about, and compare part XiJ PHILOSOPHY OF INDUCTIVE INFERENCE. 221 with part, and it could certainly draw inferences concern- ing the similarity of forms, the coexistence of qualities, or^the preponderance of a particular kind of matter in a changeless world. A solid universe, in at least approxi- mate equilibrium, is not inconceivcvble, and then the rela- tion of cause and effect would evidently be no more than the relation of before and after. As nature exists, how- ever, it is a progressive existence, ever moving and changing as time, the great independent variable, pro- ceeds. Hence it arises that we must continually compare what is happening now with what happened a moment befoi-e, and a moment before that moment, and so on, until we reach indefinite periods of past time. A comet is seen moving in the sky, or its constituent particles illumine the heavens with their tails of fire. We cannot explain the present movements of such a body without supposing its prior existence, with a definite amount of energy and a definite direction of motion ; nor can we validly suppose that our task is concluded when we find that it came wandering to our solar system through the unmeasured vastness of surrounding space. Every event must have a cause, and that cause again a cause, until wo are lost in the obscurity of the past, and are driven to the belief in one First Cause, by whom the course of nature was determined. Fallacious Use of the Term Cause. Tlie words Cause and Causation have given rise to infinite trouble and obscurity, and have in no slight degree retarded the progress of science. From the time of Aristotle, the work of philosophy has been described as the discovery of the causes of things, and Francis Bacon adopted the notion when he said " vere scire esse per causas scire." Even now it is not uncommonly supposed that the knowledge of causes is something different from other knowledge, and consists, as it were, in getting possession of the keys of nature. A singje word may thus act as a spell, and throw the clearest intellect into confusion, as I have often thought that Locke was thrown into confusion when endeavouring to find a meaning for the word power} In Mill's System of ^ Et$ay caticemtng Human Undentanding, bk. iL chap, xxi r- THE PRINCIPLES OP SCIENCE. fCHAP. 'i M I Logic the term catise seems to have re-asserted its old noxious power. Not only does Mill treat the Laws of Causation as almost coextensive with science, but he so uses the expression as to imply that when once we pass wiUun the circle of causation we deal with certainties. The philosophical danger which attaches to the use of this word may be thus described. A cause is defined as the necessary or invariable antecedent of an event so that when the cause exists the effect wiU also exist or soon follow. If then we know the cause of an event we know what will certainly happen ; and as it is implied that science, by a proper experimental method, may attain to a knowledge of causes, it follows that experience may give us a certain knowledge of future events. But nothing IS more unquestionable than that finite experience can never give us certain knowledge of the future, so that either a cause is not an invariable antecedent, or else we can never gain certain knowledge of causes. The first horn of this dilemma is hardly to be accepted. Doubtless there IS m nature some invariably acting mechanism, such that from certain fixed conditions an invariable result always emerges. But we, with our finite minds and short expenence, can never penetrate the mystery of those existences which embody the Will of the Creator and evolve it throughout time. We are in the position ot spectators who witness the productions of a compli- cated machine, but are not aUowed to examine its inti- mate structure. We learn what does happen and what does appear, but if we ask for the reason, the answer would involve an infinite depth of mystery. The simplest bit of matter, or the most trivial incident, such as the stroke of two billiard balls, offers infinitely more to learn than ever the human intellect can fathom. The word cause covers just as much untold meaning as any of the words siibstancey matter, tkougJU, existence. Confusion of Two Questions, The subject is much complicated, too, by the confusion of two distinct questions. An event having happened, we may asK—"* XI.] PHILOSOPHY OF INDUCTIVE INFERENCE. 283 (i) Is there any cause for the event ? (2) Of what kind is that cause ? No one would assert that the mind possesses any faculty capable of inferring, prior to experience, that the occurrence of a sudden noise with flame and smoke indi- cates the combustion of a black powder, formed by the mixture of black, white, and yellow powders. The greatest upholder of d priori doctrines will allow that the parti- cular aspect, shape, size, colour, texture, and other qualities of a cause must be gathered through the senses. The question whether there is any cause at all for an event, is of a totally different kind. If an explosion could happen without any prior existing conditions, it must be a new creation — a distinct addition to the universe. It may be plausibly held that we can imagine neither the creation nor annihilation of anything. As regards matter, this has long been held true ; as regards force, it is now almost universally assumed as an axiom that energy can neither come into nor go out of existence without distinct acts of Creative Will. That there exists any instinctive belief to this effect, indeed, seems doubtful. We find Lucretius, a philosopher of the utmost intellectual power and cultivation, gravely assuming that his raining atoms could turn aside from their straight paths in a self-deter- mining manner, and by this spontaneous origination of energy determine the form of the universe.^ Sir George Airy, too, seriously discussed the mathematical conditions under which a perpetual motion, that is, a perpetual source of self-created energy, might exist.* The larger part of the philosophic world has long held that in mental acts there is free will — in short, self-causation. It is in vain to attempt to reconcile this doctrine with that of an intuitive belief in causation, as Sir W. Hamilton candidly allowed. It is obvious, moreover, that to assert the existence of a cause for every event cannot do more than remove into the indefinite past the inconceivable fact and mystery of creation At any given moment matter and energy ^ De i?«rt,m NcUuraj bk. ii. IL 216-293. * Cambr%d{fe Fhiloiophieal Transaetumt (1830), yol 369—372- lu. p|». 9S4 THE PRINCIPLES OP SCIENCE. [CRAl'. I < M were equal to wliat they are at present, or they were not ; if equal, we may make the same inquiry concerning any other moment, however long prior, and we are thus obliged to accept one horn of the dilemma — existence from infinity, or creation at some moment. This is but one of the many cases in which we are compelled to believe in one or other of two alternatives, both inconceivable. My present purpose, however, is to point out that we must not confuse this supremely diflBcult question with that into which inductive science inquires on the foundation of facts. By induction we gain no certain knowledge ; but by observation, and the inverse use of deductive reasoning, we estimate the probability that an event which has occurred was preceded by conditions of specified character, or that such conditions will be followed by the event. Definition of the Term Cause, Clear definitions of the word cause have been given by several philosophers. Hobbes has said, " A cause is the sum or aggregate of all such accidents, both in the agents and the patients, as concur in the producing of the effect propounded ; all which existing together, it cannot be understood but that the effect existeth with them; or that it can possibly exist, if any of them be absent." Brown, in his Essay on Causation, gave a nearly corre- sponding statement. "A cause," he says,* "may be defined to be the object or event which immediately precedes any change, and which existing again in similar circumstances will be always immediately followed by a similar change." Of the kindred word power, he like- wise says : * *' Power is nothing more than that invariable- ness of antecedence which is implied in the belief of causation." These definitions may be accepted with the qualifica- tion that our knowledge of causes in such a sense can be probable only. The work of science consists in ascertaining the combinations in which phenomena present themselves. * Observaiumi on the Nature and Tendency of the Doctrine of Hr. Hume, concerning the Relation of Cause and Effect, Second ed. ^ 44. * Ibid. p. 97. XI.] PHILOSOPHY OF INDUCTIVE INFERENCE. 225 Concerning every event we shall have to determine its probable conditions, or the group of antecedents from which it probably follows. An antecedent is anything which exists prior to an event; a consequent is anything which exists subsequently to an antecedent. It will not usually happen that there is any probable connection between an antecedent and consequent. Thus nitrogen is an antece- dent to the lighting of a common fire ; but it is so far from being a cause of the lighting, that it renders the combustion less active. Daylight is an antecedent to all fires lighted during the day, but it probably has no appreciable effect upon their burning. But in the case of any given event it is usually possible to discover a certain number of ante- cedents which seem to be always present, and with more or less probability we conclude that when they exist the event will follow. Let it be observed that the utmost latitude is at present enjoyed in the use of the term cause. Not only may a cause be an existent thing endowed with powers, as oxygen is the cause of combustion, gunpowder the cause of explosion, but the very absence or removal of a thing may also be a cause. It is quite correct to speak of the .dryness of the Egyptian atmosphere, or the absence of moistui'e, as being the cause of tlie preservation of mummies, and other remains of antiquity. The cause of a mountain elevation, Ingleborough for instance, is the excavation of the surrounding valleys by denudation. It is not so usual to speak of the existence of a thing at one moment as the cause of its existence at the next, but to me it seems the commonest case of causation which can occur. The cause of motion of a billiard ball may be the stroke of another ball ; and recent philosophy leads us to look upon all motions and changes, a.s but so many mani- festations of prior existing energy. In all probability there is no creation of energy and no destruction, so that as regards both mechanical and molecular changes, the cause is really the manifestation of existing energy. In the same way I see not why the prior existence of matter is not also a cause as regards its subsequent existence. All science tends to show us that the existence of the universe in a particular state at one moment, is the condition of its existence at the next moment, in an apparently different f| fl 286 THE PRINCIPLES OF SCIENCE. [chap. state. When we analyse the meaning which we can attribute to the word cause^ it amounts to the existence of suitable portions of matter endowed with suitable quan-. tities of energy. If we may accept Home Tooke's asser- tion, cause has etymologically the meaning of thing hefore. Though, indeed, the origin of the word is very obscure, its derivatives, the Italian cosa, and the French chose^ mean simply thing. In the German equivalent ursache, we have plainly the original meaning of thing before, the sache denoting "interesting or important object," the English sake, and tir being the equivalent of the English «rf, he/ore. We abandon, then, both etymology and philo- sophy, when we attribute to the laws of causation any meaning beyond that of the conditions under which an event may be expected to happen, according to our observation of the previous course of nature. I have no objection to use the words cause and causation, provided they are never allowed to lead us to imagine that our knowledge of nature can attain to cer- tainty. I repeat that if a cause is an invariable and necessary condition of an event, we can never know certainly whether the cause exists or not. To us, then, a cause is not to be distinguished from the group of positive or negative conditions which, with more or less probability, precede an event. In this sense, there is no particular difference between knowledge of causes and our general knowledge of the succession of combinations, in which the phenomena of nature are presented to us, or found to occur in experimental inquiry. Distinction of Inductive and Deductive Results. We must carefully avoid confusing together inductive investigations which terminate in the establishment of general laws, and those which seem to lead directly to the knowledge of future particular events. That process only can be called induction which gives general laws, and it is by the subsequent employment of deduction that we anticipate particular events. If the observation of a number of cases shows that alloys of metals fuse at lower tempemtures than their constituent metals, I may with more or less probability draw a general inference to that El.] PHILOSOPHY OF INDUCTIVE INFERENCE. 2t7 effect, and may thence deductively ascertain the proba- bility that the next alloy examined will fuse at a lower temperature than its constituents. It has been asserted, indeed, by Mill,^ and partially admitted by Mr. Fowler,^ that we can argue directly from case to case, so that what is true of some alloys will be true of the next. Professor Bain has adopted the same view of reasoning. He thinks that Mill has extricated us from the dead lock of the syllogism and effected a total revolution in logic. He holds that reasoning from particulars to particulars is not only the usual, the most obvious and the most ready method, but that it is the type of reasoning which best discloses tJie real process.' Doubtless, this is the usual result of our reasoning, regard being had to degrees o! probability ; but these logicians fail entirely to give any explanation of the process by which we get from case to case. It may be allowed that the knowledge of future par- ticular evente is the main purpose of our investigations, and if there were any process of thought by which we could pass directly from event to event without ascending into general truths, this method would be sufficient, and certainly the briefest. It is true, also, that the laws of mental association lead the mind always to expect the like again in apparently like circumstances, and even animals of very low intelligence must have some trace of such powers of association, serving to guide them more or less correctly, in the absence of true reasoning faculties. But it is the purpose of logic, according to Mill, to ascertain whether inferences have been correctly drawn, rather than to discover them.* Even if we can, then, by habit, association, or any rude process of inference, infer the future directly from the past, it is the work of logic to analyse the conditions on which the correctness of this inference depends. Even Mill would admit that sucl analysis involves the consideration of general truths,* am' • System of LogiCf bk. II. chap. iii. • Inductive Logic, pp. 13, 14. • Bain, Deductive Logic, pp. 208, 209. • System of Logic. Introduction, § 4. Fifth ed. pp. 8, 9. • Ibid. bk. II. chap. iii. } 5, pp. 225, &c Q 2 i 'II 1 828 THE PRINCIPLES OF 8CIBN0E. [chap. in this, as iu several other impoilaut points, we might controvert Mill's own views by his own statements. It seems to me undesirable in a systematic work like this to enter into controversy at any length, or to attempt to refute the views of other logicians. But I shall feel bound to state, in a separate publication, my very deliberate opinion that many of Mill's innovations in logical science, and especially his doctrine of reasoning from particulars to particulars, are entirely groundless and false. The Grounds of Iriductive Inference. I hold that in all cases of inductive inference we must invent hypotheses, until we fall upon some hypothesis which yields deductive results in accordance with experi- ence. Such accordance renders the chosen hypothesis more or less probable, and we may then deduce, with some degree of likelihood, the nature of our future experience, on the assumption that no arbitrary change takes place in the conditions of nature. We can only argue from the past to the future, on the general principle set forth in this work, that what is true of a thing will be true of the like. So far then as one object or event differs from another, all inference is impossible, particulars as particulars can no moi-e make an infei*ence than grains of sand can make a rope. We must always rise to something which is general or same in the cases, and assuming that sameness to be extended to new cases we learn their nature. Hearing a clock tick five thousand times without exception or varia- tion, we adopt the very probable hypothesis that there is some invariably acting machine which produces those uni- form sounds, and which will, in the absience of change, go on producing them. Meeting twenty times with a bright yellow ductile substance, and finding it always to be \%vy heavy and incorrodible, I infer that there was some natural condition which tended in the creation of things to asso- ciate these properties together, and I expect to find them associated in the next instance. But there always is the possibility that some unknown change may take place between past and future cases. The clock may run down, or be subject to 'jl hundred accidents altering its condition. TLer^ is no reason in the nature of things, so far as known XI.] PHILOSOPHY OF INDUCTIVE INFERENCE. 229 to us, why yellow colour, ductility, high specific gravity, and incorrodibility, should always be associated together, and in other cases, if not in this, men's expectations have been deceived. Our inferences, therefore, always retain more or less of a hypothetical character, and are so far open to doubt. Only in proportion as our induction approximates to the character of j)erfect induction, does it approximate to certainty. The amount of uncertainty corresponds to the probability that other objects than those examined may exist and falsify our inferences ; the amount of probability corresponds to the amount of infor- mation yielded by our examination ; and the theory of probability will be needed to prevent us from over-esti- mating or under- estimating the knowledge we possess. Illustrations of the Inductive Process, To illustrate the passage from the known to the ap- parently unknown, let us suppose that the phenomena under investigation consist of numbers, and that the following six numbers being exhibited to us, we are required to infer the character of the next in the series : — 5» i5» 35, 45, 6s, 95. The question first of all arises. How may we describe this series of numbers ? What is uniformly true of them ? The reader cannot fail to perceive at the first glance that they all end in five, and the problem is, from the pi-opcr- ties of these six numbers, to infer the properties of the next number ending in five. If we test their properties by the process of perfect induction, we soon perceive that they have another common property, namely that of being divisible hyflve without remainder. May we then assert that the next number ending in five is also divisible by five, and, if so, upon what grounds ? Or extending th(^ question, Is every number ending in five divisible by five ? Does it follow that because six numbers obey a supposed law, therefore 376,685,975 or any other number, however large, obeys the law ? I answer certainly not. The law in ques- tion is undoubtedly true ; but its truth is not proved by any finite number of examples. All that these six numbers can do is to suggest to my mind the possible existence of aao THK PRINCIPLES OF SCIENCE. [OHAr. xu] PHILOSOPHY OF INDUCTIVE INFERENCE. 231 such a law ; and I then ascertain its truth, by proving deductively from the rules of decimal numeration, that any number ending in five must be made up of multiples of five, and must therefore be itself a multiple. To make this more plain, let the reader now examine the numbers — 7> 17* 37, 47, 67, 97. They all end in 7 instead of 5, and though not at equal intervals, the intervals are the same as in the previous case. After consideration, the reader will perceive that these numbers all agree in being prime numbers, or mul- tiples of unity only. May we then infer that the next, or any other number ending in 7, is a prime number? Clearly not, for on trial we tind that 27, 57, 117 are not primes. Six instances, then, treated empirically, lead us to a true and universal law in one case, and mislead us in another case. We ought, in fact, to have no confidence in any law until we have treated it deductively, and have shown that from the conditions supposed the results ex- pected must ensue. No one can show from the principles of number, that numbers ending in 7 should be primes. From the history of the theory of numbers some good examples of false induction can be adduced. Taking the following series of prime numbers, 4i,43>47, 53,61,71, 83.97, "3, 131, 151, &c., it will be found that they all agree in being values of the general expression x* + a? + 41, putting for a; in succes- sion the values, o, i, 2, 3, 4, &c. We seem always to obtain a prime number, and the induction is apparently strong, to the effect that this expression always will give piimes. Yet a few more trials disprove this false con- clusion. Put X = 40, and we obtain 40 x 40 + 40 + 41, or 41 X 41. Such a failure could never have happened, had we shown any deductive reason why a;^ + a; + 41 should give primes. There can be no doubt that what here happens with forty instances, might happen with forty thousand or forty million instances. An apparent law never once failing up to a certain point may then suddenly break down, so that inductive reasoning, as it has been described Dy some writers, can give no sure knowledge of what is to come. Babbage pointed out in his Ninth Bridge water Treatise, that a machine could be constructed to give a perfectly regular series of numbers through a vast series of steps, and yet to break the law of progression suddenly at any required point. No number of particular cases as particulars enables us to pass by inference to any new case. It is hardly needful to inquire here what can be inferred from an infinite series of facts, because they are never practically within our power ; but we may unhesitatingly accept the conclusion, that no finite number of instances can ever prove a general law, or can give us certain know- ledge of even one other instance. General mathematical theorems have indeed been dis- covered by the observation of particular cases, and may again be so discovered. We have Newton's own state- ment, to the effect that he was thus led to the all-impor- tant Binomial Theorem, the basis of the whole structure of mathematical analysis. Speaking of a certain series of terms, expressing the area of a circle or hyperbola, he says : " I reflected that the denominators were in arithmetical progression; so that only the numerical co-efficients of the numerators remained to be investigated. But these, in the alternate areas, were the figures of the powers of the number eleven, namely 11°, 11 \ ii*, ns^ ii*j that is, in the first l ; in the second I, i ; in the third i, 2, i ; in the fourth i, 3, 3, I ; in the fifth i, 4, 6, 4, i.^ I inquired, therefore, in what manner all the remaining figures could be found from the fii-st two ; and I found that if the first figure be called m, all the rest could be found by the continual multiplication of the terms of the formula «— o m- ^ m-_2 ^ m- X &a"« 3 ' 4 It is pretty evident, from this most interesting statement, that Newton, having simply observed the succession of the numbers, tried various formulae until he found one which agreed with them alL He was so little satisfied with this process, however, that he verified particular results of his new theorem by comparison with the results of common * These are the figurate numbers considered in pages 183, 187, &c ' Commercium J^istolicum. Einttola ad Oldenburgum, Oct. 24, 1676. Horsley's Works of Newton, vol. iv. p. 541. See De Morgan in Penny Cvclovcedia art ** Binofnial Theorem," p. 412. 838 THE PRINCIPLES OF SCIENCE. [CHAF. 1 < multiplication, and the rule for the extraction of the square root. Newton, in fact, gave no demonstration of his theorem ; and the greatest mathematicians of the last century, James Bernoulli, Maclaurin, Landen, Euler, Lagrange, &c., occupied themselves with discovering a con- clusive method of deductive proof. There can be no doubt that in geometiy also discoveries have been suggested by direct observation. Many of the now trivial propositions of Euclid's Elements were pro- bably thus discovered, by the ancient Greek geometers ; and we have pretty clear evidence of this in the Commen- taries of Proclus.^ Galileo was the first to examine the remarkable properties of the cycloid, the curve described by a point in the circumference of a wheel rolling on a plane. By direct observation he ascertained that the area of the curve is apparently three times that of the generating circle or wheel, but he was unable to prove this exactly, or to verify it by strict geometrical reasoning. Sir George Airy has recorded a curious case, in which he fell accidentally by trial on a new geometrical property of the sphere.* But discovery in such cases means nothing more than sugges- tion, and it is always by pure deduction that the general law is really established. As Proclus puts it, "ve must pass from sense to consideration. Given, for instance, the series of figures in the accom- panying diagram, measurement will show that the curved lines approxim te to semicircles, and the rectilinear figures to right-angled triangles. These figures may seem to suggest to the mind the general law that angles inscribed * Bk. ii. chap. iv. * PhiloMphiccU Transactions (i866), vol. 146, p. 534. \ XI.] PHILOSOPHY OF INDUCTIVE INFERENCE. 233 in semicircles are right angles ; but no number of instances, and no possible accuracy of measurement would really establish the truth of that general law. Availing ourselves of the suggestion furnished by the figures, we can only investigate deductively the consequences which flow from the definition of a circle, until we discover among them the property of containing right angles. Persons have thought that they had discovered a method of trisecting angles by plane geometrical construction, because a certain complex arrangement of lines and circles had appeared to trisect an angle in every case tried by them, and they inferred, by a supposed act of induction, that it would succeed in all other cases. De Morgan has recorded a proposed mode of trisecting the angle which could not be discriminated by the senses from a true general solution, except when it was applied to very obtuse angles.^ In all such cases, it has always turned out either that the angle was not trisected at all, or that only certain particular angles could be thus trisected. The trisectors were misled by some apparent or special coincidence, and only deductive proof could es- tablish the truth and generality of the result. In this par- ticular case, deductive proof shows that the problem attempted is impossible, and that angles generally cannot be trisected by common geometrical methods. Geometrical Reasoning. This view of the matter is strongly supported by the further consideration of geometrical reasoning. No skill and care could ever enable us to verify absolutely any one geometrical proposition. Kousseau, in his Umile, tells us that we should teach a child geometry by causing him to measure and compare figures by superposition. While a child was yet incapable of general reasoning, this would doubtless be an instructive exercise ; but it never could teach geometry, nor prove the truth of any one proposition. All our figures are rude approximations, and they may happen to seem unequal when they should be equal, and equal when they should be unequal Moreover figures may from chance be equal in case after case, and ' Budget of Paradoxes^ p. 257. li 834 THE PRINCIPLES OF SCIENCB. [CHilF yet there may be no general reason why they should be so. The results of deductive geometrical reasoning are absolutely certain, and are either exactly true or capable of bemg carried to any required degree of approximation In a perfect tnangle, the angles must be equal to one half- revolution precisely; even an infinitesimal divergence would be impossible; and I believe with equal confidence that however many are the angles of a figure, provided there are no re-entrant angles, the sum of the angles will be precisely and absolutely equal to twice as many right- angles as the figure has sides, less by four right-aufrles. In such cases, the deductive proof is absolute and com- plete ; empirical verification can at the most guard against accidental oversights. There is a second class of geometrical truths which can only be proved by approximation ; but, as the mind sees no reason why that approximation should not always go on, we arrive at complete conviction. We thus learn that the surface of a sphere is equal exactly to two-thirds of the whole surface of the circumscribing cylinder, or to four times the area of the generating circle. The area of a parabola is exactly two- thirds of that of the circumscribing parallelogram. The area of the cycloid is exactly three times that of the generating circle. These are truths that we could never ascertain, nor even verify by observation for any finite amount of difference, less than what the senses can discern, would falsify them. There are geometrical relations again which we cannot assign exactly, but can cany to any desirable degree of ap- proximation. The ratio of the circumference to the dia- meter of a circle is that of 314159265358979323846 to I, and the approximation may be carried to any ex- tent by the expenditure of sufficient labour. Mr. W Shanks has given the value of this natural constant, known as TT, to the extent of 707 places of decimals.^ Some years since, I amused myself by trying how near I could get to this ratio, by the careful use of compasses, and I did not come nearer than i part in 540. We might imagine mea- surements so accurately executed as to give us eight or ten places correctly. But the power of the hands &mi » Froeeedings 0/ the Royal Society (1872-3), yoI. xxi. p. 319. ) ^ 11.] PHILOSOPHY OP INDUCTIVE INFERENCE. 235 senses must soon stop, whereas the mental powers of de- ductive reasoning can proceed to an unlimited degree of ap- proximation. Geometrical truths, then, are incapable of verification ; and, if so, they cannot even be learnt by observation. How can I have learnt by observation a pro- position of which I cannot even prove the truth by obser- vation, when I am in possession of it ? All that observa- tion or empiiical trial can do is to suggest propositions, of which the truth may afterwards be proved deductively. If Viviani's story is to be believed, Galileo endeavoured to satisfy himself about the area of the cycloid by cutting out several large cycloids in pasteboard, and then compar- ing the areas of the curve and the generating circle by weighing them. In every trial the curve seemed to be rather less than three times the circle, so that Galileo, we are told, began to suspect that the ratio was not precisely 3 to I. It is quite clear, however, that no process of weighing or measuring could ever prove truths like these, and it remained for Torricelli to show what his master Galileo had only guessed at.^ Much has been said about the peculiar certainty of mathematical reasoning, but it is only certainty of deduc- tive reasoning, and equal certainty attaches to all correct logical deduction. If a triangle be right-angled, the squai-e on the hypothenuse will undoubtedly equal the sum of the two squares on the other sides ; but I can never be sure that a triangle is right-angled : so I can be certain that nitric acid will not dissolve gold, provided I know that the substances employed really correspond to those on which I tried the experiment previously. Here is like certainty of inference, and like doubt as to the facts. Discriminatian of Certainty and Probability, We can never recur too often to the truth that our knowledge of the laws and future events of the external world is only probable. The mind itself is quite capable of possessing certain knowledge, and it is well to discri- minate carefully between what we can and cannot know * Life of (JaliUOf Society for the Diffusion of Useful Knowledge, p. 102. 236 THE PRINCIPLES OP SCIENCE. [chap. '1 I. '. i \vith certainty. In the first place, wliatcver feeling is actually present to the mind is certainly known to that mind. If I see blue sky, I may be quite sure that I do experience the sensation of bhieness. Whatever I do feel, I do feel beyond all doubt. We are indeed very likely to confuse what we really feel with what we are inclined to associate with it, or infer inductively from it; but the whole of our consciousness, as far as it is the result of pure intuition and free from inference, is certain knowledge beyond all doubt. In the second place, we may have certainty of inference ; the fundamental laws of thought, and the rule of substitution (p. 9), are cei-tainly true ; and if my senses could inform me that A was indistinguishable in colour from B, and B from C, then I should be equally cerfain that A was indistinguish- able from C. In short, whatever tnith there is in the premises, I can certainly embody in their correct logical result. But the certainty generally assumes a hypothetical character. I never can be quite sure that two colours are exactly alike, that two magnitudes are exactly equal, or that two bodies whatsoever are identical even in their apparent qualities. Almost all our judgments involve quantitative relations, and, as will be shown in succeeding chaptei-s, we can never attain exactness and certainty where continuous quantity enters. Judgments concerning discontinuous quantity or numbers, however, allow of cer- tainty ; I may establish beyond doubt, for instance, that the difference of the squares of ly and 13 is the product of 17 + 13 and 17—13, and is therefore 30 x 4, or 120. Inferences which we draw concerning natural objects are never certain except in a hypothetical point of view. It might seem to be certain that iron is magnetic, or that gold is incapable of solution in nitric acid ; but, if we carefully investigate the meanings of these state- ments, they will be found to involve no certainty but that of consciousness and that of hypothetical inference. For what do I mean by iron or gold? If I choose a remarkable piece of yellow substance, call it gold, and then immerse it in a liquid which I call nitric acid, and find that there is no change called solution, then conscious- ness has certainly informed me that, with my meaning of the terms, '* Grold is insoluble in nitric acid." I may further mh- XL] PHILOSOPHY OP INDUCTIVE INFERENCR 237 be certain of something else ; for if this gold and nitric acid remain what they were, 1 may be sure there will be no solution on again trying the experiment. If I take other portions of gold and nitric acid, and am sure that they really are identic«d in properties with the former portions, I can be certain that there will be no solution. But at this point my knowledge becomes purely hypothetical ; for how can I be sure without trial that the gold and acid are really identical in nature with what I formerly called gold and nitric acid. How do I know gold when I see it ? If 1 judge by the apparent qualities — coloui-, ductility, specific gravity, &c., I may be misled, because there may always exist a substance which to the colour, ductility, specific gravity, and other specified qualities, joins others which we do not expect Similarly, if iron is magnetic, as shown by an experiment with objects answering to those names, then all iron is magnetic, meaning all pieces of matter identical with my assumed piece. But in trying to identify iron, I am always open to mistake. Nor is this liability to mis- take a matter of speculation only.^ The history of chemistry shows that the most confident inferences may have been falsified by the confusion of one substance with another. Thus strontia was never discri- minated from baryta until Klapruth and Hauy detected differences between some of their properties. Accordingly chemists must often have inferred concerning strontia what was only true of baryta, and vice versd. There is now no doubt that the recently discovered substances, caesium and rubidium, were long mistaken for potassium.^ Other elements have often been confused together — for instance, tantalum and niobium ; sulphur and selenium ; cerium, lanthanum, and didymium ; yttrium and erbium. Even the best known laws of physical science do not exclude false inference. No law of nature has been better established than that of universal gravitation, and we believe with the utmost confidence that any body capable of affecting the senses will attract other bodies, and fall to the earth if not prevented. Euler remarks m 1 Professor Bowen has excellently stated this view. Logic Cambridge, U.S.A., 1866, p. 354. ^ BoBcoe's Spectrum Anuiysis, ist edit., p. 98. Treatise on ■sai^AB '( 838 THE PRINCIPLES OF SCTENCB. [CBIP. tliat, although he had never made trial of the stones which compose the church of Magdeburg, yet he had not the least doubt that all of them were heavy, and would fall if unsupported. But he adds, that it would be extremely difficult to give any satisfactory explanation of this confident belief.^ The fact is, that the belief ought not to amount to certainty until the experiment has been tried, and in the meantime a slight amount of uncer- tainty enters, because we cannot be sure that the stones of the Magdeburg Church resemble other stones in all their properties. In like manner, not one of the inductive truths which men have established, or think they have established, is really safe from exception or reversal. Lavoisier, when laying the foundations of chemistry, met with so many instances tending to show the existence of oxygen in all acids, that he adopted a general conclusion to that effect, and devised the name oxygen accordingly. He entertained no appreciable doubt that the acid existing in sea salt also contained oxygen;* yet subsequent ex- perience falsified his expectations. This instance refers to a science in its infancy, speaking relatively to the possible achievements of men. But all sciences are and ever will remain in their infancy, relatively to the extent and complexity of the universe which they undertake to investigate. Euler expresses no more than the truth when he says that it would be impossible to fix on any one thing really existing, of which we could have so perfect a know- ledge as to put us beyond the reach of mistake." We may be quite certain that a comet will go on moving in a similar path if all circumstances remain the same as before ; but if we leave out this extensive qualification, our predictions will always be subject to the chance of falsification by some unexpected event, such as the division of Biela's comet or the interference of an unknown gravi- tating body. xu] PHILOSOPHY OF INDUCTIVE INFERENCE. 239 Inductive inference might attain to certainty if our knowledge of the agents existing throughout the universe were complete, and if we were at the same time certain tliat the same Power which created the universe would allow it to proceed without arbitrary change. There is always a possibility of causes being in existence without our knowledge, and these may at any moment produce an unexpected effect. Even when by the theory of pro- babilities we succeed in forming some notion of the com- parative confidence with which we should receive in- ductive results, it yet appears to me that we must make an assumption. Events come out like balls from the vast ballot-box of nature, and close observation wiU enable us to form some notion, as we shall see in the next chapter, of the contents of that ballot-box. But we must still assume that, between the time of an observation and that to which our infei-ences relate, no change in the ballot-box has been made. * Euler's Letten to a Oerman Priw«M, translated by Hunter. 2nd ed., vol. ii. pp. 17, 18. 2 Lavoisier's Chemistry ^ translated by Kerr. 3id ed., pp. X14, lai, 123. * Euler's Letters, vol. il p. 21. I ) CHAPTER XII. THE IHDUCTIVE Oil INVERSE APPLICATION OF TUB I'UEOKT 01* PfiOBABlLITY. (' We have hitherto considered the theory of probability only in its simple deductive employment, in which it enables us to determine from given conditions the probable character of events happening under those conditions. But as deductive reasoning when inversely applied con- stitutes the process of induction, so the calculation of probabilities may be inversely applied ; from the known character of certain events we may argue backwards to the probability of a certain law or condition governing those events. Having satisfactorily accomplished this work, we may indeed calculate forwards to the probable character of future events happening under the same con- ditions ; but this part of the process is a direct use of deductive reasoning (p. 226). Now it is highly instructive to find that whether the theoiy of probability be deductively or inductively ap- plied, the calculation is always performed according to the principles and rules of deduction. The probability that an event has a particular condition entirely depends upon the probability that if the condition existed the event would follow. If we 'take up a pack of common playing cards, and observe that they are arranged in per- fect numerical order, we conclude beyond all reasonable doubt that they have been thus intentionally arranged by some person acquainted with the usual order of sequence. This conclusion is quite irresistible, and rightly m min. CH. XII.] THE INDUCITVE OR INVERSE METHOD. 241 80 ; for there are but two suppositions which we can make as to the reason of the cards being in that particular order : — ^ 1. They may have been intentionally arranged by some one who would probably prefer the numerical order. 2. Ihey may have fallen into that order by chance, that is, by some series of conditions which, being unknown to us cannot be known to lead by preference to the particular order m question. The latter supposition is by no means absurd, for any one order is as likely as any other when there is no prepon. demting tendency. But we can readily calculate by the doctrine of permutations the probability that fifty-two objects would fall by chance into any one particular order. Filty-two objects can he arranged in 52 x 51 x . . x 3 X 2 X I or about 8066 x (io)«* possible orders, the number obtained requiring 6S plaoes of figures for its lull expression. Hence it is excessively unlikely that anyone should ever meet with a pack of cards arrancred in perfect order by accident. If we do meet with a pack so arranged, we inevitably adopt the other supposi- tion, that some person, haWng reasons for preferring that special order, has thus put them together. We know that of the immense number of possible orders the numerical order is the most remarkable ; it is useful as proving the perfect constitution of the paok, and It is the intentional result of certain games. At any rate the probability that intention should produce that order is incompai-ably greater than the probability that chance should produce it ; and as a certain pack exists in that order, we rightly prefer the supposition which most pro- oably leads to the observed result By a similar mode of reasoning we every day arrive and validly arrive, at conclusions approximating to cer- tainty. Whenever we observe a perfect resemblance between two objects, as, for instance, two printed pages two engravings, two coins, two foot-prints, we are war- i^nted m asserting that they proceed from the same type the same plat«, the same pair of dies, or the same boot' Ana why ? Because it is almost impossible that with amerent types, plates, dies, or boots some apparent dis- tinction of form should not be produced. It is impossible ^^mm V l\ f HI f 242 THE PHINCIPLBS OP SCIENCE. [cnAF for the hand of the most skilful artist to make two objecte alike, so that mechanical repetition is the only probable explanation of exact similarity. We can often establish with extreme probability that one document is copied from another. Suppose that each document contains io,cxx> words, and that tlie same word is incorrectly spelt in each. There is then a probability of less than i in io,cx» that the same mistake should be made in each. If we meet with a second error occurring in each document, the probability is less than i in 10,000 X 9999' that two such coincidences should occur by chance, and the numbers grow with extreme rapidity for more numerous coincidences. We cannot make any precise calculations without taking into account the character of the errors committed, concerning the conditions of which we have no accurate means of estimating probabilities. Nevertheless, abundant evidence may thus be obtained as to the derivation of documents from each other. In the examination of many sets of logarithmic tables, six remarkable errors were found to be present in all but two, and it was proved that tables printed at Paris, Berlin, Florence, Avignon, and even in China, besides thirteen sets printed in England between the years 1633 and 1822, were derived directly or indirectly from some common source.^ With a certain amount of labour, it is possible to establish beyond reasonable doubt the relationship or genealogy of any number of copies of one document, pro- ceeding possibly from parent copies now lost The rela- tions between the manuscripts of the New Testament have been elaborately investigated in this manner, and the same work has been performed for many classical writings, especially by German scholars. Principle of the Inverse Method, The inverse application of the rules of probability entirely depends upon a proposition which may be thus stated, nearly in the words of Laplace.* If an event can ^ Tiardner, Edinburgh Review, July 1834, p. 277. ■ Mimoires par divert Savons, torn, vl ; quoted by Todhunter in his Hiitory of the Theory of FrobabHity, p. 458. xii.] THE INDUCTIVE OR INVERSE METHOD. 243 he produced by any one of a ceHain number of different catise^, all eqmlly probable a priori, the probabilities of the existence of tliese causes as infeired from the event, are pro- portional to tlie probabilities of the event as derived from these causes. In other words, the most probable cause of an event which has happened is that which would most pro- bably lead to the event supposing the cause to exist; but all other possible causes are also to be taken into account with probabilities proportional to the probability that the event would happen if the cause existed. Suppose, to fix our ideas clearly, that E is the event, and C, Cj C3 are the three only conceivable causes. If C exist, the probability is pi that E would follow ; if Cj or Cj exist, the like pro- babilities are respectively p^ and p^ Then as ;?j is to p^, so is the probability of Cj being the actual cause to the probabQity of 0, being it ; and, similarly, as p^ is to p., so 13 the probability of C, being the actual cause to the probability of Cj being it By a simple mathematical pro- cess we arrive at the conclusion that the ** ^ual probability of Cj being the cause is Pi + Pt + Pi* and the similar probabilities of the existence of C, and C3 are, * , ^« and ^ Pi-tPt-hPi Pi+Pi+Pi The sum of these three fractions amounts to unity, which correctly expresses the certainty that one cause or other must be in operation. We may thus state the result in general language. If it is certain that one or other of the supposed cames exists, the probability that any one does exist is the proba- bility that if it exists the event happens, divided by the mm of all the similar probabilities. Tlierfe may seem to be an mtncacy in this subject which may prove distasteful to some readers ; but this intricacy is essential to the subject in hand. No one can possibly understand the principles of inductive reasoning, unless he wiU take the trouble to master the meaning of this rule, by which we recede from an event to the probability of each of its possible causes. This rule or principle of the indirect method is that which common sense leads us to adopt almost instinctively, R 2 II • ( 144 THE PRINCIPLES OF SCIENCE. [OBAP. before we have any comprehension of the principle in its general form. It is easy to see, too, that it is the rule 'which will, out of a great multitude of cases, lead us most often to the truth, since the most probable cause of an event really means that cause which in the greatest number of cases produces the event Donkin and Boole have given demonstrations of this principle, but the one most easy to comprehend is that of Poisson. He imagines each possible cause of an event to be represented by a distinct ballot-box, containing black jind white balls, in such a ratio that the probability of a white ball being drawn is equal to that of the event happening. He further supposes that each box, as is possible, contains the same total number of balls, black and white ; then, mixing all the contents of the boxes together, he shows that if a white ball be drawn from the aggregate ballot-box thus formed, the probability that it proceeded from any par- ticular ballot-box is represented by the number of white balls in that particular box, divided by the total number of white balls in all the boxes. This result corresponds to that given by the principle in question.^ Thus, if there be three boxes, each containing ten balls in all, and respectively containing seven, four, and three white balls, then on mixing all the balls together we have fourteen white ones ; and if we draw a white ball, that is if the event happens, the probability that it came out of 7 the first box is J^ ; which is exactly equal to , . V t" s ' Tff + TTF + TTF the fmction given by the rule of the Inverse Method. Simple Applications of the Inverse Method. In many cases of scientific induction we may apply the principle of the inverse method in a simple manner. If only two, or at the most a few hypotheses, may be made as to the origin of certain phenomena, we may sometimes easily calculate the respective probabilities. It was thus that Bunsen and Kirchlioff established, with a probability ittle short of certainty, that iron exists in the sun. On comparing the spectra of sunlight and of the light proceed- I Poiason, lUeherchu iur la ProbabilUe da JugemcuU, Paiia, 1837, W^ 82, 83. XII.] THE INDUCTIVE OR INVERSE METHOD. S45 ing from the incandescent vapour of iron, it became appa- rent that at least sixty bright lines in the spectrum of iron coincided with dark lines in the sun's spectrum. Such coin- cidences could never be observed with certainty, because, even if the lines only closely approached, the instrumental imperfections of the spectroscope would make them appa- rently coincident, and if one line came within half a milli- metre of another, on the map of the spectra, they could not be pronounced distinct. Now the average distance of the solar Imes on Kirchhofif's map is 2 mm., and if we throw down a line, as it were, by pure chance on such a map, the probability is about one-half that the new line will fall within J mm. on one side or the other of some one of the solar lines. To put it in another way, we may suppose that each solar line, either on account of its real breadth, or the defects of the instrument, possesses a breadth of i mm., and that each line in the iron spectrum has a like breadth. The probability then is just one-half that the centre of each iron line will come by chance within i mm. of the centre of a solar line, so as to appear to coincide with it The probability of casual coincidence of each iron line with a solar line is in like manner i. Coinci- dence in the case of each of the sixty iron lines is a very unlikely event if it arises casually, for it would have a probability of only {^)^ or less than i in a trHlion. The odds, in short, are more than a million million millions to umty against such casual coincidence.^ But on the other hypothesis, that iron exists in the sun, it is highly probable that such coincidences would be observed ; it is immensely more probable that sixty coincidences would be observed if iron existed in the sun, than that they .should arise from chance. Hence by our principle it is immensely probable that iron does exist in the sun, AH the other interesting results, given by the comparison of spectra, rest upon the same principle of probability. Ihe almost complete coincidence between the spectra of Bolar, lunar, and planetary light renders it pracMcally certain that the light is all of solar origin, and is reflected trom the surfaces of the moon and planets, suffering onh 1 ! ,^»rchhoff's Researches <m the Solar Spectrum. M»tod by Roanoe, pp. 18, 19. Fiist part, trans- !i«sa 846 THE PKINCIPLES OF SCIENCE. [chap. li' Blight alteration from the atmospheres of some of tlie planets. A fresh confirmation of the truth of the Coper- nican theory is thus furnished. Herschel proved in this way the connection between the direction of the oblique faces of quartz crystals, and the direction in which the same crystals rotate the plane of polarisation of light. For if it is found in a second crystal that tlie relation is the same as in the first, the probability of this happening by chance is J ; the probability that in another crystal also the direction will be the same is i, and so on. The probability that in n 4- I crystals there would be casual agi-eement of direc- tion is the nth power of i. Thus, if in examining fourteen crystals the same relation of the two phenomena is dis- covered in each, the odds that it proceeds from uniform conditions are more than 8000 to i.^ Since the first observations on this subject were made in 1820, no excep- tions have been observed, so that the probability of in- variable connection is incalculably great. It is exceedingly probable that the ancient Egyptians had exactly recorded the eclipses occurring during long periods of time, for Diogenes Laertius mentions that 373 solar and 832 lunar eclipses had been observed, and the ratio between these numbers exactly expresses that which would hold true of the eclipses of any long period, of say 1200 or 1300 years, as estimated on astronomical grounds. It is evident that an agreement between small numbers, or customary numbers, such as seven, one hundred, a myriad, &c., is much more likely to happen from chance, and therefore gives much less presumption of de- pendence. If two ancient writers spoke of the sacrifice of oxen, they would in all probability describe it as a heca* tomb, and there would be nothing remarkable in the coin- cidence. But it is impossible i;o point out any special reason why an old writer should select such numbers as 373 and 832, unless they had been the results of observa- tion. On similar grounds, we must inevitably believe in the ' Edinburgh Review^ No. 185, vol. xcii. July 1850, p. 32 ; Herschel's , p. 421 ; Transixctioii* of the Cambridge I'hilosophical iiodcty, E'lmys vuJ. i. JI.43. XII.] THE INDUCTIVE OR INVERSE METHOD. 247 human origin of the flint flakes so copiously discovered of late years. For though the accidental stroke of one stone against another may often produce flakes, such as are occasionally found on the sea-shore, yet when several flakes are found in close company, and each one bears evidence, not of a single blow only, but of several suc- cessive blows, all conducing to form a symmetrical knife- like form, the probability of a natural and accidental origin becomes incredibly small, and the contrary suppo- sition, that they are the work of intelligent beings, approximately certain.^ The TJieory of Prohahility in Astronomy, The science of astronomy, occupied with the simple relations of distance, magnitude, and motion of the heavenly bodies, admits more easily than almost any other science of interesting conclusions founded on the theory of probability. More than a century ago, in 1767, Michell showed the extreme probability of bonds connecting together systems of stai-s. He was struck by the unexpected number of fixed stars which have companions close to them. Such a conjunction mi^^ht happen casually by one star, although possibly at' a great distance from the other, happening to Lie on a straight line passing near the earth. But the probabilities are so greatly against such an optical union happening often in the expanse of the heavens, that Michell asserted the existence of some connection between most of the double stai's. It has since been estimated by Struve, that the odds are 9570 to i against any two stars of not less than the seventh magnitude falling within the appa- rent distance of four seconds of each other by chance, and yet ninety-one such cases were known when the estimation was made, and many more cases have since been discovered. There were also four known triple stars, and yet the odds against the appearance of any one such conjunction are ' 73*524 to I.* The conclusions of Michell have been * Evans* Ancient Stone Implementt of Great Britain. London, 1872 (Longmans). ^Herschel, Outliius of Astronomy, 1849, p. 565 ; but ToJlhunter, in his Hittory of the Theory of Probability, p. 335, states that the calculations do not agree with those published by Struve. I 1} y. S48 THE PRINCIPLES OF SCIENCE. [chap. entirely verified by the discovery that many double stars are connected by gravitation. Michell also investigated the probability that the six brightest stars in the Pleiades should have come by accidents into such striking proximity. Estimating the number of stars of equal or greater brightness at 1500, he found the odds to be nearly 500,000 to i against casual conjunction. Extending the same kind of argument to other clusters, such as that of Pi-msepe, the nebula in the hilt of Perseus* sword, he says:^ "We may with the highest probability conclude, tho odds against the contrary opinion being many million millions to one, that the stars are really collected together in clusters in some places, where they form a kind of system, while in others there are either few or none of them, to whatever cause this may !!>e owing, whether to their mutual gravitation, or to some other law or appointment of the Creator." The calculations of Michell have been called in question by the late James D. Forbes,^ and ^Ir. Todhunter vaguely countenances his objections,' otherwise I should not have thought them of much weight. Certainly Laplace accepts Michell's views,* and if Michell be in error it is in the methods of calculation, not in the general validity of his reasoning and conclusions. Similar calculations might no doubt be applied to the peculiar drifting motions which have been detected by Mr. R A. Proctor in some of the constellations.* The odds are veiy greatly against any numerous group of stars mov- ing together in any one direction by chance. On like grounds, there can be no doubt that the sun has a con- siderable proper motion because on the average the fixed btars show a tendency to move apparently from one point of the heavens towards that diametrically opposite. The sun's motion in the contrary direction would explain this tendency, otherwise we must believe that thousands of stars accidentally agree in their direction of motion, or are * Philoiophical TrantactionSf 1767, vol Ivii p. 431. ' PhilosophiccU Magazinty 3rd Senee, voL xxxvii. p. 401, December i8qo ; also August 1849. ^trtory, &c., p. 334. * Euai FhUosophique, p. 57. Proceedings of the Royal Society y 20 January, 1870 ; Philosophical aiagazine^ 4th Series, vol. xxxix. p. 381. XII.] THE INDUCTIVE OR INVERSE METHOD. 240 urged by some common force from which the sun is exempt. It may be said that the rotation of the earth is proved in like manner, because it is immensely more pro- bable that one body would revolve than that the sun moon, planets, comets, and the whole of the stars of the heavens should be whiried round the earth daily, with a uniform motion superadded to their own peculiar motions. This appears to be mainly the reason which led Gilbert one of the eariiest English Copemicans. and in every way an admirable physicist, to admit the rotation of the earth while Francis Bacon denied it In contemplating the planetary system, we are struck with the similarity in direction of nearly all its movements Newton remarked upon the regularity and uniformity of these motions, and contrasted them with the eccentricity and irregularity of the cometary orbits.^ Could we in fact, look down upon the system from the northern side we should see all the planets moving round from west to east, the satellites moving round their primaries, and the sun planets, and satellites rotating in the same direction, with some exceptions on the verge of the system. In the time of Laplace eleven planets were known, and the direc- tions of rotation were known for the sun, six planets the satellites of Jupiter, Saturn's ring, and one of his satellites Ihus there were altogether 43 motions all concurrin^r namely : — ^* Orbital motions of eleven planets . . 1 1 Orbital motions of eighteen satellites . .18 Axial rotations ! 14. 43 The probabiHty that 43 motions independent of each other would coincide by chance is the 42nd power of i, so that the odds are about 4,400.000,000,000 to i in favour of some common cause for the uniformity of direction. This probability, as Laplace obsei-ves,2 is higher than that of many historical events which we undoubtingly believe In the present day, the probability is much increased by the discovery of additional planets, and the rotation of other ! -Pnnctpta, bk. ii. General scholium- S,tu^"l* ^^*Vo«op;it^, p. 55. Laplace appears to count the rings of S'ltum as giving two independent movemeiite. ^ f« ■I S50 THE PRINCIPLES OF SCIENCE. I [chap. satellites, and it is only slightly weakened by the fact that some of the outlying satellites are exceptional in direction, there being considerable evidence of an accidental dis- turbance in the more distant parts of the system. Hardly less remarkable than the uniform direction of motion is the near approximation of the orbits of the planets to a common plane. Daniel Bernoulli roughly estimated the probability of such an agreement arising from accident as l -5- (12)® the greatest inclination of any orbit to the sun's equator being I-I2th part of a quadrant. Laplace devoted to this subject some of his most ingenious investigations. He found the probability that the sum of the inclinations of the planetary orbits would not exceed by accident the actual amount (•914187 of a right angle' for the ten planets known in 1801) to be (^^y (9 14 187),*** or about •00000011235. This probability may be com- bined with that derived from the direction of motion, and it then becomes immensely probable that the constitution of the planetary system arose out of uniform conditions, or, as we say, from some common cause.^ If the same kind of calculation be applied to the orbits of comets, the result is very different.' Of the orbits which have been determined 48*9 per cent, only are direct or in the same direction as the planetary motions.* Hence it becomes apparent that comets do not properly belong to the solar system, and it is probable that they are stray portions of nebulous matter which have accidentally become attached to the system by the attractive powers* of the sun or Jupiter. The General Inverse Problem, In the instances described in the preceding sections, we have been occupied in receding from the occurrence of certain similar events to the probability that there > Lubbock, Essay on Frobability, p. 14. De Morgan, Encye. Metrap. art. Probability ^ p. 412. Tod hunter's History of the Theory of Probabilityj p. 543. Concerning the objections raised to these conclusions by Boole, see the Philosophical Magazine, 4tb Series, vol. ii. p. 98. Boole's Latos of Thought^ pp. 364-375, 2 Laplace, Essai Philosophiaue^ pp. 55, 56. > Chambers* Asironomy, 2nd ed. pp. 346-40, I W I i XII.] THE INDUCTIVE OR INVERSE METHOD. IBt must have been a condition or cause for such events. We have found that the theory of probability, although never yielding a certain result, often enables us to establish an hypothesis beyond the reach of reasonable doubt. There is, however, another method of applying the theory, which possesses for us even greater interest, because it illustrates, in the most complete manner, the theory of inference adopted in this work, which theory indeed it suggested. The problem to be solved is as follows : — An event Jtaving liappened a certain mimher of times, and failed a certain number of times, required the pro- lability tJuit it wUl happen any given number of times in the future under the same circumstances. AH the larger planets hitherto discovered move in one direction round the sun ; what is the probability that, if a new planet exterior to Neptune be discovered, it will move in the same direction ? All known permanent gases, ex- cept chlorine, are colourless ; what is the probability that, if some new permanent gas should be discovered, it will be colourless ? In the general solution of this problem, we wish to infer the future happening of any event from' the number of times that it has already been observ^ed to happen. Now, it is very instructive to find that there is no known process by which we can pass directly from the data to the conclusion. It is always requisite to recede from the data to the probability of some hypothesis, and to make that hypothesis the ground of our inference concerning future events. Mathematicians, in fact, make every hypothesis which is applicable to the question in hand ; they then calculate, by the inverse method, the probability of every such hypothesis according to the data, and the probability that if each hypothesis be true, the required future event will happen. The total pro- bability that the event will happen is the sum of the separate probabilities contributed by each distinct hypo- thesis. To illustrate more precisely the method of solving the problem, it is desii-able to adopt some concrete mode of representation, and the ballot-box, so often employed by mathematicians, will best serve our purposa Let the happening of any event be represented by the drawing of a white tall from a ballot-box, while the fiedlure of an (I -, M It III- 852 THE PRINCIPLES OF SCIENCE. [on A p. event is represented by the drawing of a black ball. Now, in the inductive problem we are supposed to bo ignorant of the contents of the ballot-box, and are required to ground all our inferences on our experience of those con- tents as shown in successive drawings. Rude common sense would guide us nearly to a tnie conclusion. Thus, if we had drawn twenty balls one after another, replacing the ball after each drawing, and the ball had in each case proved to be white, we should believe that there was a considerable preponderance of white balls in the urn, and a probability in favour of drawing a white ball on the next occasion. Though we had drawn white balls for thousands of times without fail, it would still be possible that some black balls lurked in the urn and would at last appear, so that our inferences could never be certain. On the other hand, if black balls came at intervals, we should expect that after a certain number of trials the black balls would appear again from time to time with somewhat the same frequency. The mathematical solution of the question consists in little more than a close analysis of the mode in which our common sense proceeds. If twenty white balls have been drawn and no black ball, my common sense tells me that any hypothesis which makes the black balls in the urn considerable compared with the white ones is improbable ; a preponderance of white balls is a more probable hypo- thesis, and as a deduction from this more probable hypo- thesis, I expect a recurrence of white balls. The mathe- matician merely reduces this process of thought to exact numbers. Taking, for instance, the hypothesis that there are 99 white and one black ball in the urn, he can calcu- late the probability that 20 white balls would be drawn in succession in those circumstances; he thus forms a definite estimate of the probability of this hypothesis, and knowing at the same time the probability of a white ball reappearing if such be the contents of the urn, he com- bines these probabilities, and obtains an exact estimate that a white hall will recur in consequence of this hypo- thesis. But as this hypothesis is only one out of many possible ones, since the ratio of white and black balls may be 98 to 2, or 97 to 3, or 96 to 4, and so on, he has to repeat the estimate for every such possible hypothesis. XIL] THE INDUCTIVE OR INVERSE METHOD. To make the method of solving the problem perfectly evident, I will describe in the next section a very simple case of the problem, originally devised for the purpose by Condorcet, which was also adopted by Lacroix,i and has passed into the works of De Morgan, Lubbock, and others. Simple IlludrcUion of the Inverse Problem, Suppose it to be known that a ballot-box contains only four black or white balls, the ratio of black and white balls being unknown. Four drawings having been made with replacement, and a white ball having appeared on each occasion but one, it is required to determine the proba- bility that a white ball will appear next time. Now the hypotheses which can be made as to the contents of the urn are very limited in number, and are at most the following five : — 4 white and o black balls 3 n n I n » 2 n » 2 >» 1 I »» >» 3 »> n ^ »» » 4 » » The actual occurrence of black and white balls in the drawings puts the first and last hypothesis out of the question, so that we have only three left to consider. If the box contains three white and one black, the probability of drawing a white each time is }, and a black i ; so that the compound event observed, namely, three white and one black, has the probability J X } X | x J, by the rule already giveu (p. 204). But as it is indifferent in what order the balls are drawn, and the black ball might come first, second, third, or fourth, we must multi- ply by four, to obtain the probability of three white and one black in any order, thus getting JJ. Taking the next hypothesis of two white and two black balls in the urn, we obtain for the same proba- bility the quantity J x J x J x J x 4, or ^J, and from the thmi hypothesis of one white and three black we deduce likewise i x i x J x J x 4, or ^. According, then, as we * Traite iUnuntaire du Caleul det ProbabiliUi, 3rd ed. 08^^^ !>. 148. ^ ^^^* \) < ; M Ill ^wSy li fi S54 THE PRINCIPLES OF SCIENCE. [crap. adopt the first, second, or third hypothesis, the proba- bility that the result actually noticed would follow is ||, J4, and ^^. Now it is certain that one or other of these hypotheses must be the true one, and their absolute probabilities are proportional to the probabilities that the observed events would follow from them (pp. 242, 243). All we have to do, then, in order to obtain the absolute pro- bability of each hypothesis, is to alter these fractions in a uniform ratio, so that their sum shall be unity, the expression of certainty. Now, since 27 + 16 + 3 = 46, this will be effected by dividing each fraction by 46, and multiplying by 64. Thus the probabilities of the first, second, and third hypotheses are respectively — 27 16 3 46' 46* 46' The inductive part of the problem is completed, since we have found that the urn most likely contains three white and one black ball, and have assigned the exact probability of each possible supposition. But we are now in a position to resume deductive reasoning, and infer the probability that the next drawing will yield, say a white ball. For if the box contains three white and one black ball, the pro- bability of drawing a white one is certainly J ; and as the probability of the box being so constituted is JJ, the com- pound probability that the box will be so filled and will give a white ball at the next trial, is 27 3 81 -^ X 7 or - . 46 184 Again, the probability is jj that the box contains two white and two black, and under those conditions the probability is J that a white ball will appear ; hence the probability that a white ball will appear in consequence of that condition, is 16 ^ I 32 56 ^ 5 **' 184* From the third supposition we get in like manner the probability Since one and not more than one hypothesis can be true. XII. ] THE INDUCTIVE OR INVERSE METHOD. 255 we may add together these separate probabilities, and we find that «! , 32 , 3 116 184 "^ 184 ■*■ i"8i *''' T8i IS the complete probability that a white ball will be next drawn under the conditions and data supposed. Gemral SohUion of the Inverse Problem, In the instance of the inverse method described in the last section, the balls supposed to be in the ballot-box were few, for the purpose of simplifying the calculation. m order that our solution may apply to natural phe- nomena we must render our hypotheses as little arbitrary as possible. Having no d priori knowledge of the con- ditions of the phenomena in question, there is no limit to the variety of hypotheses which might be suggested. Mathematicians have therefore had recourse to the most extensive suppositions which can be made, namely, that the ballot-box contains an infinite number of balls- they have then varied the proportion of white to black balls continuously, froni the smallest to the greatest possible proportion, and estimated the aggregate probability which results from this comprehensive supposition. To explain their procedure, let us imagine that, instead of an infinite number, the ballot-box contains a large finite number of balls, say 1000. Then the number of white balls might be I or 2 or 3 or 4, and so on, up to 999. Supposing that three white and one black ball have been drawn from the urn as before, there is a certain very small probability that this would have occurred in the case of a box containing one white and 990 black balls ; there is also a smaU probability that from such a ^\ l-f-..^®'^* Y^ ^^^^^ ^ w^i^- Compound these probabdities, and we have the probability that the next ball really will bo white, in consequence of the existence of that proportion of baUs. If there be two white and ogS black balls m the box, the probabOity is greater and will increase until the balls are supposed to be in the proper- tion of tho^ drawn. Now 999 different hypotheses are possible, and the calculation is to be made for each of tnese, and their aggregate taken as the final result. It is Ill; 2&6 THE PRINCIPLES OP SCIENCE. [CHAF. apparent that as the number of balls in the box is increased, the absolute probability of any one hypothesis concerning the exact proportion of balls is decreased, but the aggregate results of all the hypotheses will assume the character of a wider average. - x,- When we take the step of supposing the balls withm the urn to be infinite in number, the possible proportions of white and black balls also become infinite, and the probability of any one proportion actually existing is infinitely small. Hence the final result that the next ball drawn will be white is really the sum of an infinite number of infinitely small quantities. It might seem impossible to calculate out a problem having an infinite number of hypotheses, but the wonderful resources of the integral calculus enable this to be done with far greater facility than if we supposed any large finite number of balls, and then actually computed the results. I will not attemp.t to describe the processes by which Laplace finally accomplished the complete solution of the problem. They are to be found described in several English works, espe- cially De Morgan's Treatise on Probabilities, in the Encij- dopcedia Metropolitana, and Mr. Todhunter's History of the Tluory of Probability. The abbreviating power of mathematical analysis was never more strikingly shown. But I may add that though the integral calculus is employed as a means of summing infinitely numerous results, we in no way abandon the principles of com- binations already treated. We calculate the values of infinitely numerous factorials, not, however, obtaining their actual products, which would lead to an infinite number of figures, but obtaining the final answer to the problem by devices which can only be comprehended after study of the integiul calculus. It must be allowed that the hypothesis adopted by Laplace is in some degree arbitrary, so that there was some opening for the doubt which Boole has cast upon it.^ But it° may be replied, (i) tliat the supposition of an infinite number of balls treated in the manner of Laplace is less arbitrary and more comprehensive than any other that can be suggested. (2) The result does not differ > Law of Thought, pp. 368-375« xiL] THE INDUCTIVE OR INVERSE METHOD. 867 much from that which would be obtained on the hypothesis of any large finite number of balls. (3) The supposition leads to a series of simple formulas which can be applied with ea^e m many cases, and which bear aU the appearance ot truth so far as it can be independently judged by a sound and practiced understanding. Rules of the Inverse Method, By the solution of the problem, as described in the last section, we obtain the following series of simple rules I. To find th^ probabUity that an event which has not hjiherto been observed to fail will happen once more, dimde the nurriber of times the event has been observed increased by one, bi the same number increased by two If there have been m occasions on which a certain event might have been observed to happen, and it has happened on all those occasions, then the probabiHty that it will happen on the next occasion of the same kind is ^^LdLl. For instance, we may say that there are nine pla'^e^ in tlie planetary system where planets might exist obeying Bodes law of distance, and in every place there is a planet obeying the law more or less exactly, althouch no reason is known for the coincidence. Hence the probability that the next planet beyond Neptune will conform to the law is |f . 2. To find the probaklUy that an event which has not hUherto failed will not fail for a certain number of new occasions, divide the number of times the event has hav^ V^ increased by one, by th^ same number increased bv »neand the nurriber of times it is to happen. An event having happened m times without fail, the probabiHty that it will happen n more times is **+' ^us the probability that three new planets w^iJd^obey ^de s law is « ; but it must be aUowed that this, as weU for fL?TT ^^lt>,wo^ld be much weakened by the fact that Neptune can barely be said to obey the law numht^^T''^ I^^"^. ^PP^^ «^ A^ a certain ICZff'^'J''^'^.^^ ^o6a^% that ii will happen W(5 r^ tim^, divide the n.uwher of times the eoe^ithas 8 258 THE PRINCIPLES OF SCIENCE. [CBAP. xilJ the INDUCTIVE OR INVERSE METHOD. r It ill 269 hajrpened increased hy one, hy the whole numher of times the event has happened or failed in>creased hy tvx). If an event has happened m times and failed n times, the probability that it will happen on the next occasion ia — !!!Li-L_. Thus, if we assume that of the elements dis- covered up to the year 1873, 50 are metallic and 14 non- metallic, then the probability that the next element dis- covered will be metallic is ^. Again, since of 37 metals which have been sufficiently examined only four, namely, sodium, potassium, lanthanum, and lithium, are of less density than water, the probability that the next metal examined or discovered wiU be less dense than water is --^— j- — or -* . 37 + 2 39 We may state the results of the method in a more general manner thus,^ — If under given circumstances cer- tain events A, B, C, &c., have happened respectively m, n, p, &c., times, and one or other of these events must happen, then the probabilities of these events are propor- tional to m + I, n + I, i> + I, Ac, so that the probability of A will be m-\- I But if new wi-|-i+n-fi-|-p-|-i-|-&c. events may happen in addition to those which have been observed, we must assign unity for the probability of such new event. The odds then become i for a new event, m + I for A, n + I for B, and so on, and the absolute probability of A is — ; , **, — ; r-5— • It is interesting to trace out the variations of probability according to these rules. The first time a casual event happens it is 2 to i that it will happen again ; if it does happen it is 3 to i that it will happen a third time ; and on successive occasions of the like kind the odds become 4, 5^ 6, &c., to I. The odds of course will be discriminated from the probabilities which are successively }, }, |, &c. Thus on the first occasion on which a person sees a shark, and notices that it is accompanied by a little pilot fish, the odds are 2 to i, or the probability }, that the next shark will be so accompanied. * De Morgan's Eisaiy on Probabilities, Cabinet CycIoptediA. p. 67. When an event has happened a very great number of times. Its happenmg once again approaches nearly to cer- ^!S^y- .. ^® suppose the sun to have risen one thousand million times, the probability that it wiU rise again, on the ground of this knowledge merely, is '»Qoo,ooo,ooq 4- i But then the probability that it will conSZTistfotL long a period in the future is only '>«)o,ooo,ooo + 1 ^^ ^^^^ exactly i The probabiHty that itQ'^Se'so rising a thousand times as long is only about ^^. The lesson which we may draw from these figures is'^te that which we n^v^ Jfj Z""^^? ^?^^'' ^^"^^ly^ ^^^^ experience Tr^Zlf^w w"^ knowledge, and that it is exceedingly mprobable that events mil always happen as we observe them. Inferences pushed far beyond their data soon lose any considerable probability. De Morgan has 8aid,i " No fimte experience whatsoever can justify us in saying that nr fW I^ '^"^ comcide with the past'inall time to come, or that there is any probabihty for such a conclusion." On tiie other hand, we gain the assurance that experience k^^!. ^ t^^f'^ ^^^ P'^^^^^^ ^ill give ^is the n^Sf ^^^?;^^v«"t« with an unUmited degree of subject to arbitrary interference onfv Zh\^ "-"^"l^ understood that these probabilities are only such as arise from the mere happening of the events 3^W%f '"^ ^T^^'^^ ^^^^^' ^-- other%Turts concerning those evente or the general laws of nature. AU our knowledge of nature is indeed founded in like manner upon observation, and is therefore only probable. The law of gravitation itself is only probably true BiS when a number of different facts, observed under ?he most — circumstances, are found to be harmonized under a supposed law of nature, the probability of the law approxi- mates dosely to certainty. ^Each science rests upoHo 31^""^'^ ^^^;. *°^ ^"^^^ «^ °^^<^^ ««PP«rt from nnalogies or connections with other sciences, that ther^ D^b^nr r'^^ ^''^. T^^ ^^^^ ^^^ judgment of the probability of an event depends entirely upoi a few ante- » EMtay on Probabilitiet, p. 128. s 2 THE PRINCIPLES OP SOIENOB. (oHAr. U I I! Il cedent events, disconnected from the general body of physical science. Events, again, may often exhibit a regularity of suc- cession or preponderance of character, which the simple formula will not take into account. For instance, the majority of the elements recently discovered are metals, so that the probability of the next discovery being that of . a metal, is doubtless greater than we calculated (p. 258). At the more distant parts of the planetary system, there are symptoms of disturbance which woidd prevent our placing much reliance on any inference from the prevailing order of the known planets to those undiscovered ones which may possibly exist at great distances. These and all like complications in no way invalidate the theoretic truth of the fonnuhis, but render their sound application much more difficult. Erroneous objections have been raised to the theory of probability, on the ground that we ought not to trust to our d priori conceptions of what is likely to happen, but should always endeavour to obtain precise experimental data to guide us.^ This course, however, is perfectly in accordance with the theory, which is our best and only guide, whatever data we possess. We ought to be always applying the inverse method of probabilities so as to take into account all additional information. When we throw up a coin for the first time, we are probably quite ignorant whether it tends more to fall head or tail upwards, and we must therefore assume the probability of each event as ^. But if it shows head in the firat throw, we now have very slight experimental evidence in favour of a tendency to show head. The chance of two heads is now slightly greater than J, which it appeared to be at first,* and as we go on throwing the coin time after time, the probability of head appearing next time constantly varies in a slight degree according to the character of our previous experienca As Laplace remarks, we ought always to have regard to such considerations in common life. Events when closely scrutinized will hardly ever prove to be quite independent, and the slightest pre- * J. S. Mill, System of Logte, 5th edition, bk. iii. chap, xviii. § 3. ^ Todliuuter's Uiitoryy pp. 472, 598 xn.] THB INDUCTIVE OR INVERSE METHOD. 261 ponderance one way or the other is some evidence of connection, and m the absence of better evidence should be taken into account. The grand object of seeking to estimate the probabUity of future events from past experience, seems to have beon entertained by James Bernoulli and De Moivre, at least such w^ the opinion of Condorcet ; and BernouUi may be said to have solved one case of the problem.^ The English wntei-s Bayes and Price are, however, undoubtedly the tet who put forward any distinct rules on the subject 2 Condorcet and several other eminent mathematicians ad- vanced the mathematical theory of the subject : but it was reserved to tlie immortal Laplace to bring to the subject the fuU power of Iils genius, and carry the solution of the problem almost t« perfectioa It is instructive to observe that a theoiy which arose from petty games of chance, the rules and the very names of which are forgotten, m-ad nail v advanced, until it embraced the most sublime problems of science and finally undertook to measure the value and certainty of all our inductions. Fortuitous Coincidences, We should have studied the theory of probability to very little purpose, if we thought that it would furnish us with an infallible guide. The theory itself points out the approximate certainty, that we shall sometimes De deceived by extraordinary fortuitous coincidences, ihere 13 no run of luck so extreme that it may not wt?^"' ^""^ ^^ ""^^ ^^PP^^ *o ^' or in our time, as well as to other persons or in other times. We may be lorced by correct calculation to refer such coincidence? w a necessary cause, and yet we may be deceived. All l.ti,'.?^?'^''^ ""^ probability pretends to give, is ihs resuit %n the long run, as it is caUed, and this really means Tnw? *'V^''*^^ ?^ "^^^ ^^^g ^^y finite experience, tlofiy®' long, chances may be against us. Nevertheless thfl 7 '^ *^^ ^^* S^^^^ ^0 can have. If we always tmnk and act according to its well-interpreted indications, J paS!1^^* ?^' pp- 378. 379. THE PRINCIPLES OF SCIENCE. [chap. r ' 1) i < .'■ H we shall have the best chance of escaping error ; and if all persons, throughout all time to come, obey the theory in like manner, they will undoubtedly thereby reap the greatest advantaga No rule can be given for discriminating between coincidences which are casual and those which are the effects of law. By a fortuitous or casual coincidence, we mean an agreement between events, which nevertheless arise from wholly independent and different causes or con- ditions, and which will not always so agree. It is a fortuitous coincidence, if a penny thrown up repeatedly in various ways always falls on the same side ; but it would not be fortuitous if there were any similarity in the motions of the hand, and the height of the thi-ow, so as to cause or tend to cause a uniform result. Now among the infinitely numerous events, objects, or relations in the universe, it is quite likely that we shall occasionally notice casual coincidences. There are seven intervals in the octave, and there is nothing very improbable in the colours of the spectrum happening to be apparently divisible into the same or similar series of seven intervals. It is hardly yet decided whether this apparent coincidence, with which Newton was much struck, is well founded or not,^ but the question will probably be decided in the negative. It is certainly a casual coincidence which the ancienU noticed between the seven vowels, the seven strings of the lyre, the seven Pleiades, and the seven chiefs at Thebes.' The accidents connected with the number seven have mis- led the human intellect throughout the historical period. Pythagoras imagined a connection between the seven planets and the seven intervals of the mouochord. The alchemists were never tired of drawing inferences from the coincidence in numbers of the seven planets and the seven metals, not to speak of the seven days of the week. A singular circumstance was pointed out concerning the dimensions of the earth, sun, and moon; the sun*6 diameter was almost exactly no times as great as the > Newton's Opticks, Bk. I., Part il Prop. 3 ; Nature, toL l p 286 * Axutotle's MetaphysieSf xiil 6. 3. XIA.] THE INDUCTIV E OR INVERSE METHOD. 263 earth's diameter, while in almost exactly the same ratio the mean distance of the earth was greater than the sun's diameter, and the mean distance of the moon from the earth was greater than the moon's diameter. The agree- ment was so close that it might have proved more than casual, but its fortuitous character is now sufficiently shown by the fact, that the coincidence ceases to be remarkable when we adopt the amended dimensions of the planetary system. A considerable number of the elements have atomic weights, which are apparently exact multiples of that of hydrogen. If this be not a law to be ultimately ex- tended to all the elements, as supposed by Prout, it is a most remarkable coincidence. But, as I have observed we have no means of absolutely discriminating accidental coincidences from those which imply a deep producing cause. A coincidence must either be very strong in itself, or it must be corroborated by some explanation or connection with other laws of nature. Little attention was ever given to the coincidence concerning the dimen- sions of the sun, earth, and moon, because it was not very strong in itself, and had no apparent connection with the principles of physical astronomy. Prout's Law bears more probability because it would bring the constitution of the elements themselves in close connection with the atomic theory, representing them as built up out of a simpler substance. In historical and social matters, coincidences are fre- quently pointed out which are due to chance, although there is always a strong popular tendency to regard them as the work of design, or as having some hidden meaning. If to 1794, the number of the year in which Robespierre fell, we add the sum of its digits, the result is 181 5, the year in which Napoleon fell ; the repetition of the process gives 1830 the year in which Charles the Tenth abdicated. Again, the French Chamber of Deputies, in 1830, consisted of 402 members, of whom 221 formed the party called "La queue de Robespierre," while th^ .emainder, 181 in number, were named " Les honn^tes gens." If we give to each letter a numerical value corres/'-^nding to its place in the alphabet, it will be found that tlie sum of the values of the letters in each name exactly indicates the number of the party. l> '! ( ill it M4 THE PRINCIPLES OF SCIENCE. [chap. A number of such coincidences, often of a very curious chamcter, might be adduced, and the probability against the occurrence of each is enormously great. They must be attributed to chance, because they cannot be shown to have the slightest connection with the general laws of nature ; but persons are often found to be greatly in- fluenced by such coincidences, regarding them as evidence of fatality, that is of a system of causation governing human afTairs independently of the ordinary laws of nature. Let it be remembered that there are an infinite number of opportunities in life for some strange coincidence to pre- sent itself, so that it is quite to be expected that remark- able conjunctions will sometimes happen. In all matters of judicial evidence, we must bear in mind the probable occurrence from time to time of un- accountable coincidences. The Roman jurists refused for this reason to invalidate a testamentary deed, the wit- nesses of which had sealed it with the same seal. For witnesses independently using their own seals might be found to possess identical ones by accident^ It is well known that circumstantial evidence of apparently over- whelming completeness will sometimes lead to a mistaken judgment, and as absolute certainty is never really attain- able, every court must act upon probabilities of a high amount, and in a certain small proportion of cases they must almost of necessity condemn the innocent victims of a remarkable conjuncture of circumstances.* Popular judgments usually turn upon probabilities of far less amount, as when the palace of Nicomedia, and even the bedchamber of Diocletian, having been on fire twice within fifteen days, the people entirely refused to believe that it could be the result of accident. The Romans believed that there was fatality connected with the name of Sextus. ** Semper sub Sextis perdita Roma fuiL" The utmost precautions will not provide against all contingencies. To avoid errors in important calculations, 1 Possiint autem omnes testes et uno annalo signare testamentum Qi H enim si septem anmili una sculptura fuerint, secundum quod Pomponio visum est ? — Justinian^ ii. tit. x. 6. * See Wills on CircuTnttantial Evidence n. ia8. XII.] THE INDUCTIVE OR INVERSE METHOD. 265 it is usual to have them repeated by different computers ; but a case is on record in which three computers made exactly the same calculations of the place of a star, and yet all did it wrong in precisely the same manner, for no apparent reason.^ Summary of the Theory of Inductive Inference. Tlie theoiy of inductive inference stated in this and the previous chapters, was suggested by the study of the Inverse Method of Probability, but it also bears much resemblance to the so-called Deductive Method described by Mill, in his celebrated System of Logic. Mill's views concerning the Deductive Method, probably form the most original and valuable part of his treatise, and I should have ascribed the doctrine entirely to him, had I not found that the opinions put forward in other parts of his work are entirely inconsistent with the theory here upheld. As this subject is the most important and difficult one with which we have to deal, I will try to remedy the imperfect manner in which I have treated it, by giving a recapitulation of the views adopted. All inductive reasoning is but the inverse application of deductive reasoning. Being in possession of certain particular facts or events expressed in propositions, we imagme some more general proposition expressing the existence of a law or cause; and, deducing the particular results of that supposed general proposition, we observe whether they agree with the facts in question. Hypo- thesis is thus always employed, consciously or unconsci- ously. The sole conditions to which we need conform in framing any hypothesis is, that we both have and exercise the power of inferring deductively from the hypothesis to the particukr results, which are to be compared with the known facts. Thus there are but three steps in the process of induction : — (i) Framing some hypothesis as to the character of the general law. (2) Deducing consequences from that law. * MemoirioftJie EovcU Astronomical Society, vol iv. p. 200, quoted by Lardner, Edinburgh Review, July 1834, p. 278. ! u i ^ i 266 THE PRINCIPLES OF SCIENCE. [OHAi-. xii.] THE INDUCTIVE OR INVERSE METHOD. 267 It I! ij' (3) Observing whether the consequences agree with the particular facts under consideration. In very simple cases of inverse reasoning, hypothesis may seem altogether needless. To take numbers again as a convenient illustration, I have only to look at the series, I, 2, 4, 8, 16, 32, &c., to know at once that the general law is that of geo- metrical progression ; I need no successive trial of various hypotheses, because I am familiar with the series, and have long since learnt from what general formula it proceeds. In the same way a mathematician becomes acquainted with the integrals of a number of common formulas, so that he need not go through any process of discovery. But it is none the less true that whenever previous reason- ing does not furnish the knowledge, hypotheses must be framed and tried (p. 124). There naturally arise two cases, according as the nature of the subject admits of certain or only probable deductive reasoning. Certainty, indeed, is but a singular case of probability, and the general principles of procedure are always the same. Nevertheless, when certainty of infer- ence is possible, the process is simplified. Of several mutually inconsistent hypotheses, the results of which can be certainly compared with fact, but one hypothesis can ultimately be entertained. Thus in the inverse logical problem, two logically distinct conditions could not yield the same series of possible combinations. Accordingly, in the case of two terms we had to choose one of six different kinds of propositions (p. 136), and in the case of three terms, our choice lay among 192 possible distinct hypotheses (p. 140). Natural laws, however, are often quantitative in character, and the possible hypotheses are then infinite in variety. When deduction is certain, comparison with fact is needed only to assure ourselves that we have rightly selected the hypothetical conditions. The law establishes itself, and no number of particular verifications can add to its probability. Having once deduced from the prin- ciples of algebra that the difference of the squares of two numbers is equal to the product of their sum and dif- ference, no number of particular trials of its truth will render it more certain. On the other hand, no finite number of particular verifications of a supposed law will render that law certain. In short, certainty belongs only to the deductive process, and to the teachings of direct intuition ; and as the conditions of nature are not given by intuition, we can only be certain that we have got a correct hypotliesis when, out of a limited number con- ceivably possible, we select that one which alone agrees with the facts to be explained. In geometry and kindred branches of mathematics, deductive reasoning is conspicuously certain, and it would often seem as if the consideration of a single diagram yields us certain knowledge of a general proposition. . But in reality all this certainty is of a purely hypothetical character. Doubtless if we could ascertain that a sup- posed circle was a true and perfect circle, we could be certain concerning a multitude of its geometrical pro- perties. But geometrical figures are physical objects, and the senses can never assure us as to their exact forms. The figures really treated in Euclid's Mements are imaginary, and we never can verify in practice the conclusions which we draw with certainty in inference; questions of degree and probability enter. Passing now to subjects in which deduction is only probable, it ceases to be possible to adopt one hypothesis to the exclusion of the others. We must entertain at the same time all conceivable hypotheses, and regard each with the degree of esteem proportionate to its probability. We go through the same steps as before. (1) We frame an hypothesis. (2) We deduce the probability of various series of pos- sible consequences. (3) We compare the consequences with the particular facts, and observe the probability that such facts would happen under the hypothesis. The above processes must be performed for every con- ceivable hypothesis, and then the absolute probability of each will be yielded by the principle of the inverse method (p. 242). As in the case of certainty we accept that hypothesis which certainly gives the required results, so now we accept as most probable that hypothesis which most probably gives the results; but we are obliged to entwrtain at the some time all other hypotheses with l\i i I If (/ S68 THE PRINCIPLES OF SCIENCE. [OHAr I: degrees of probability proportionate to the probabilities that they would give the same results. So far we have treated only of the process by which we pass from special facts to general laws, that inverse application of deduction which constitutes induction. But the direct employment of deduction is often com- bined with the inverse. No sooner have we established a general law, than the mind rapidly draws particular consequences from it In geometry we may almost seem to infer that because one equilateral triangle is equiangular, therefore another is so. In reality it is not because one is' that another is, but because all are. The geometrical con- ditions are perfectly general, and by what is sometimes called parity of reasoning whatever is true of one equilateral triangle, so far as it is equilateral, is true of all equilateral triangles. Similarly, in all other cases of inductive inference, where we seem to pass from some particular instances to a new instance, we go through the same process. We form an hypothesis as to the logical conditions under which the given instances might occur; we calculate inversely the probability of that hypothesis, and com- pounding this with the probability that a new instance would proceed from the same conditions, we gain the absolute probability of occurrence of the new instance in virtue of this hypothesis. But as several, or many, or even an. infinite number of mutually inconsistent hypo- theses may be possible, we must repeat the calculation for each such conceivable hypothesis, and then the complete probability of the future instance will be the sum of the separate probabilities. The complication of this process is often very much reduced in practice, owing to the fact that one hypothesis may be almost certainly true, and other hypotheses, though conceivable, may be so im- probable as to be neglected without appreciable error. When we possess no knowledge whatever of the con- ditions from which the events proceed, we may be unable to form any probable hypotheses as to their mode of origin. We have now to fall back upon the general solution of the problem effected by Laplace, which consists in admitting on an equal footing every conceivable ratio of favourable and unfavourable diances for the production XIL] THE INDUCTIVE OR INVERSE METHOD. 269 of the event, and then accepting the aggregate result as the best which can be obtained This solution is only to be accepted in the absence of all better means, but like other results of the calculus of probability, it comes to our aid where knowledge is at an end and ignorance begins and It prevents us from over-estimating the knowledge we possess. The general results of the solution are in accord- ance with common sense, namely, that the more often an event has happened the more probable, as a general rule IS Its subsequent recurrence. With the extension of experience this probabiHty increases, but at the same time the probability is slight that events will long continue to happen as they have previously happened. We have now pursued the theory of inductive inference as far as can be done with regard to simple logical or numencal relations. The laws of nature deal with time and space, which are infinitely divisible. As we passed from pure logic to numerical logic, so we must now pass from questions of discontinuous, to questions of continuous quantity, encountering fresh considerations of much dif- ficulty. Before, therefore, we consider how the great in- ductions and generalisations of physical science illustrate the views of mductive reasoning just explained, we must break off for a time, and review the means which we possess of measuring and comparing magnitudes of time space mass, force, momentum, energy, and the various manifestations of energy in motion, heat, electricity, chemical change, and the other phenomena of nature -11 r' I 'f i I » :* ' ; I BOOK III. METHODS OF MEASUREMENT. CHAPTER XIII. II! if THE EXACT MEASUREMENT OP PHENOMENA As physical science advances, it becomes more and more accurately quantitative. Questions of simple logical . fact after a time resolve themselves into questions of degree, time, distance, or weight. Forces hardly suspected to exist by one generation, are clearly recognised by the next, and precisely measured by the third generation, , But one condition of this rapid advance is the invention of suitable instruments of measurement. We need what Francis Bacon called InstarUias citantes, or evocantes, methods of rendering minute phenomena perceptible to the senses ; and we also require Instantice radii or curri- evliy that is measuring instruments. Accordingly, the introduction of a new instrument often forms an epoch in the history of science. As Davy said, " Nothing tends so much to the advancement of knowledge as the application of a new instrument. The native intellectual powers of men in different times are not so much the causes of the different success of their labours, as the peculiar nature of the means and artificial resources in their possession." In the absence indeed of advanced theory and analyti OH. XIII.] MEASUREMENT OP PHENOMBJ^A 271 cal power, a very precise instrument would be useless. Measuring apparatus and mathematical theory should didi- vmcQ pari passu, and with just such precision as the theorist can anticipate results, the experimentalist should be able to compare them with experience. The scrupulously accurate observations of Flamsteed were the proper complement to the intense mathematical powers of Newton. Every branch of knowledge commences with quantita- tive notions of a very rude character. After we have far progressed, it is often amusing to look back into the infancy of the science, and contrast present with past methods. At Greenwich Observatory in the present day, the hundredth part of a second is not thought an in- considerable portion of time. The ancient Chaldeans recorded an eclipse to the nearest hour, and the early Alexandrian astronomers thought it superfluous to dis- tmguish between the edge and centre of the sun. By the introduction of the astrolabe, Ptolemy and the latei Alexandrian astronomers could determine the places of the heavenly bodies within about ten minutes of arc Little progress then ensued for thirteen centuries, until Tycho Brahe made the first great step towards accuracy, not only by employing better instruments, but even more by ceasing to regard an instrument as correct Tycho, in fact, determined the errors of his instruments, and corrected his observations. He also took notice' of the effects of atmospheric refraction, and succeeded m attaining an accuracy often sixty times as great as that of Ptolemy. Yet Tycho and Hevelius often erred several minutes in the determination of a starts place, and It was a great achievement of Roemer and Flamsteed to reduce this error to seconds. Bradley, the modern Hip- parchus, carried on the improvement, his errors in right ascension, according to Bessel, being under one second of time, and those of declination under four seconds of arc. In the present day the average error of a single observa- tion is probably reduced to the half or quarter of what it was in Bradley's time; and further extreme accuracy is ^t^^^^ ^y ^e multiplication of observations, and their skilful combination according to the theory of error. Some of the more important constants, for instance that \ «« i i i> ' w III ,1 ^■'( H^H .' rM ) , ^^1 I \ 1 1, :■) it II S72 THE PRINOrPLBS OP SCIENCE. [chap. of nutation, have been determined within the tenth part of a second of space.^ It would be a matter of great interest to trace out the dependence of this progress upon the introduction of new instruments. The astrolabe of Ptolemy, the tele- scope of Galileo, the pendulum of Galileo and Huyghens, the micrometer of Horrocks, and the telescopic sights and micrometer of Gascoygne and Picard, Kcemer's transit in- strument, Newton's and Hadley's quadrant, Dollond's achromatic lenses, Harrison's chronometer, and Ramsden's dividing engine — such were some of the principal addi- tions to astronomical apparatus. The result is, that we now take note of quantities, 300,000 or 400,000 times as small as in the time of the Chaldseaus. It would be interesting again to compare the scrupulous accuracy of a modem trigonometrical survey with Erato- sthenes' rude but ingenious guess at the difference of lati- tude between Alexandria and Syene — or with Norwood's measurement of a degree of latitude iu 1635. " Sometimes I measured, sometimes I paced," said Norwood ; " and I believe I am within a scantling of the truth." Such was the germ of those elaborate geodesical measurements which have made the dimensions of the globe known to us within a few hundred yards. In other branches of science, the invention of an instru- ment has usually marked, if it has not made, an epoch. The science of heat might be said to commence with the construction of the thermometer, and it has recently been advanced by the introduction of the thermo-electric pile. Chemistry has been created chiefly by the careful use of the balance, •which forms a unique instance of an instru- ment remaining substantially in the form in which it was first applied to scientific purposes by Archimedes. The balance never has been and probably never can be im- proved, except in details of construction. The torsion balance, introduced by Coulomb towards thf* end of last century, has rapidly become essential in many branches of investigation. In the hands of Cavendish and Baily, it gave a determination of the earth's density ; applied in the galvanometer, it gave a delicate measure of electrical * Baily, British Association Catalogue of Stars, pp. 7, 23. ziil] MEASUREMENT OF PHENOMENA. 873 forces, and is indispensable in the thermo-electric pDa This balance is made by simply suspending any light rod by a thin \vire or thread attached to the middle point. And we owe to it almost all the more delicate investiga- tions in the theories of heat, electricity, and magnetism." Though we can now take note of the millionth of an inch in space, and the millionth of a second in time, we must not overiook the fact that in other operations of science we are yet in the position of the Chaldteans. Not many years have elapsed since the magnitudes of the stars, meaning the amounts of light they send to the observer's eye, were guessed at in the rudest manner, and the astronomer adjudged a star to this or that order of magnitude by a rough comparison with other stars of the same order. To Sir John Herschel we owe an attempt to introduce a uniform method of measurement and expression, bearing some relation to the real photometric magnitudes of the stars.^ Previous to the researches of Bunsen and Roscoe on the chemical action of light, we were devoid of any mode of measuring the energy of light ; even now the methods are tedious, and it is not clear that they give the energy of light so much as one of its special effects. Many natural phenomena have hardly yet been made the subject of measurement at all, such as the intensity of sound, the phenomena of taste and smell, the magnitude of atoms, the temperature of the electric spark or of the sun's photosphere. To suppose, then, that quantitative science ti-eats only of exactly measurable quantities, is a gross if it be a common mistake. Whenever we are treating of an event which either happens altogether or does not happen at all, we are engaged with a non-quantitative phenomenon, a matter of fact, not of degree ; but whenever a thing may be greater or less, or twice or thrice as great as another, whenever, in short, ratio enters even in the rudest manner, there science will have a quantitative character. There can be Uttle doubt, indeed, that every science as it pro- firesses will become gradually more and moio quantita- tive. Numerical precision is the soul of science, as * Outlines of Astronomy, 4th ed. sect. 781, p. 522 Observations at the Cape of Good Hope, &c., p. 371 Results of T In (J 'U It 274 THE PRINCIPLES OF SCIENCE. [OBAF. Herschel said, and as all natural objects exist in flpace, afid involve molecular movements, measurable in velocity and extent, there is no apparent limit to the ultimate extension of quantitative science. But the reader must not for a moment suppose that, because we depend more and more upon mathematical methods, we leave logical methods behind us. Number, as I have endeavoured to show, is logical in its origin, and quantity is but a development of number, or analogous thereto. Division of the Subject, The genei-al subject of quantitative investigation will have to be divided into several parts. We shall firstly consider the means at our disposal for measuring phe- nomena, and thus rendering them more or less amenable to mathematical treatment This task will involve an analysis of the principles on which accui-ate methods of measurement are founded, forming the subject of the remainder of the present chapter. As measurement, how- ever, only yields ratios, we have in the next chapter to consider the establishment of unit magnitudes, in terms of which our results may be expressed. As every pheno- menon is usually the sum of several distinct quantities depending upon different causes, we have next to investi- gate in Cliapter XV. the methods by which we may disen- tangle complicated effects, and refer each part of the joint effect to its separate cause. It yet remains for us in subsequent chapters to treat of quantitative induction, properly so called. We must follow out the inverse logical method, as it presents itself in problems of a far higher degree of difficulty than those which treat of objects related in a simple logical manner, and incapable of merging into each other by addition and subtraction. Cowlinuous Quantity, The phenomena of nature are for the most part mani- fested in quantities which increase or decrease continu- ously. When we inquire into the precise meaning of continuous quantity, we find that it can only be described uii.] MEASUREMENT OP PHENOMENA. 275 as that which is divisible without limit. We can divide a millimetre mto ten, or a hundred, or a thousand, or ten thousand parts, and mentally at any rate we can carry on the division ad infinUum. Any finite space, then must be conceived as made up of an infinite number of parts each infinitely small. We cannot entertain the simplest geometrical notions without allowing this The conception of a square involves the conception of *a side and diagonal, which, as Euclid beautifully proves in the 117th proposition of his tenth book, have no common measure,! meaning no finite common measure. Incom- mensurable quantities are, in fact, those which have for their only common measure an infinitely small quantity It is somewhat startling to find, too, that in theory incommen- surable quantities will be infinitely more frequent than commensurable. Let any two lines be drawn haphazard ; It is infinitely unlikely that they will be commensurable so that the commensurable quantities, which we are sup- posed to deal with in practice, are but singular cases among an infinitely greater number of incommensurable cases. Practically, however, we ti-eat all quantities as made up of the least quantities which our senses, assisted by the best measunng instruments, can perceive. So long as microscopes were uninvented, it was sufficient to regard an inch as made up of a thousand thousandths of an inch; now we must treat it as composed of a million mmionths. We might apparently avoid all mention of mlmitely small quantities, by never carrying our approxi- mations beyond quantities which the senses can appreciate In geometry, as thus treated, we should never assert two quantities to be equal, but only to be apparently equal. c\ Legendre really adopts this mode of treatment in the twentieth proposition of the first book of his Geometry • and It is practically adopted throughout the physical sciences, as we shall afterwards see. But though our nngers, and senses, and instruments must stop somewhere, there is no reason why the mind should not go on. W« can see that a proof which is only carried through a few Bteps in fact, might be carried on without limit, and it i« • Sec De Moi^gan, Study of Mathematics, in V.K.Q. Library, p. 8s T 2 i. / '■ \ :i 'I i I 11 h w •l I 276 THE PRINCIPLES OF SCIENCE. [crap this consciousness of no stopping-place, which renders Euclid's proof of his 117th proposition so impressive. Try how we will to circumvent the matter, we cannot really avoid the consideration of the infinitely small and the infinitely great. The same methods of approximation which seem confined to the finite, mentally extend them- selves to the infinite. One result of these considerations is, that we cannot possibly adjust two quantities in absolute equality. The suspension of Mahomet's coffin between two precisely equal magnets is theoretically conceivable but practically impossible. The story of the Merchant of Venice turns upon the infinite improbability that an exact quantity of liesh could be cut. Unstable equilibrium cannot exist in nature, for it is that which is destroyed by an infinitely small displacement. It might be possible to balance an egg on its end practically, because no egg has a surface of perfect curvature. Suppose the egg shell to be perfectly smooth, and the feat would become impossible. T/ie Fallacious Indications of tlu Senses. I may briefly remind the reader how little we can trust to our unassisted senses in estimating the degree or magnitude of any phenomenon. The eye cannot correctly estimate the comparative brightness of two luminous bodies which differ much in brilliancy ; for we know that the iris is constantly adjusting itself to the intensity of the light received, and thus admits more or less light according to circumstances. Tlie moon which shines vnth almost dazzling brightness by night, is pale and nearly imperceptible while the eye is yet affected by the vastly more powerful light of day. Much has been recorded concerning the compamtive brightness of the zodiacal light at different times, but it would be difficult to prove that these changes are not due to the varying darkness at the time, or the different acuteness of the observer's eye. For a like reason it is exceedingly difficult to esta- blish the existence of any change in the form or compara- tive brightness of nebulae; the appearance of a nebula greatly depends upon the keenness of sight of the observer, or the accidental condition of freshness or XIIL.] MEASUREMENT OF PHENOMENA. 277 fatigue of his eya The same is true of lunar obser- vations; and even the use of the best telescope fails to remove this difficulty. In judging of colours, again, we must remember that light of any given colour tends to dull the sensibility of the eye for light of the same colour. Nor is the eye when unassisted by instruments a much better judge of magnitude. Our estimates of the size of minute bright points, such as the fixed stars, are com- pletely falsified by the effects of irradiation. Tycho calculated from the apparent size of the star-discs, that no one of the principal fixed stars could be contained within the area of the earth's orbit. Apart, however, from irradiation or other distinct causes of error our visual estimates of sizes and shapes are often astonishingly incorrect Artists almost invariably dmw distant moun- tains in ludicrous disproportion to nearer objects, as a comparison of a sketch with a photograph at once shows. The extraordinary apparent difference of size of the sun or moon, according as it is high in the heavens or near the horizon, should be sufficient to make us cautious in accepting the plainest indications of our senses, unassisted by instrumental measurement As to statements concern- ing the height of the aurora and the distance of meteors, they are to be utterly distrusted. When Captain Parry says that a ray of the aurora shot suddenly downwards between him and the (land which was only 3,000 yards distant, we must consider him subject to an illusion of sense. ^ It is true that errors of observation are more often errors of judgment than of sense. That which is actually seen must be so far truly seen ; and if we correctly interpret the meaning of the phenomenon, there would be no error at all But the weakness of the bare senses as measuring instruments, arises from the fact that they import varying conditions of unknown amount, and we cannot make the requisite corrections and allowances as in the case of a solid and invariable instrument Bacon has excellently stated the insufficiency of the ' I««>mi8, On th€ Aurora Borealis, Smithsonian Transactiona. quoting Parry's Third Voyage, p. 61. I ' . '■ /I I ■ III J I 978 THE PRINCIPLES OF SCIENCE. [chap. senses for estimating the magnitudes of objects, or de- tecting the degrees in which phenomena present them- selves. " Tilings escape the senses/' he says, " because the object is not sufficient in quantity to strike the sense : as all minute bodies ; because the percussion of the object is too great to be endured by the senses: as the form of the sun when looking directly at it in mid-day ; because the time is not proportionate to actuate the sense: as the motion of a bullet in the air, or the quick circular motion of a firebrand, which are too fast, or the hour-hand of a common clock, which is too slow ; from the distance of the object as to place: as the size of the celestial bodies, and the size and nature of all distant bodies; from prepossession by another object : as one powerful smell renders other smells in the same room imper- ceptible ; from the interruption of interposing bodies : as the internal parts of animals ; and because the object is unfit to make an impression upon the sense : as the air or the invisible and untangible spirit which is in- cluded in every living body." Complexity of Quantitative Questions. One remark which we may well make in entering upon quantitative questions, has regard to the great variety and extent of phenomena presented to our notica So long as we deal only with a simply logical question, that question is merely, Does a certain event happen ? or. Does a certain object exist ? No sooner do we regard the event or object as capable of more and less, than the question branches out into many. We must now ask, How much 18 It compared with ite cause ? Does it change when the amount of the cause changes ? If so, does it change in the same or opposite direction ? Is the change in simple proportion to that of the cause ? If not, what more com- plex law of connection holds true ? This law determined satisfactorily in one series of circumstances may be varied under new conditions, and the most complex relations of several quantities may ultimately be established. In every question of physical science there is thus a series of steps the first one or two of which are usually TPftde with ease while the succeeding ones demand more XIII.] MEASUREMENT OF PHENOMENA. S70 and more careful measurement. We cannot lay down any invariable series of questions which must be asked from nature. The exact character of the questions will vary according to the nature of the case, but they will usually be of an evident kind, and we may readily illus- trate them by examples. Suppose that we are investigat- ing the solution of some salt in water. The first is a purely logical question : Is there solution, or is there not ? Assuming the answer to be in the affirmative, we next inquire, Does the solubility vary with the temperature, or not ? In all probability some variation will exist, and we must have an answer to the further question. Does the quantity dissolved increase, or does it diminish with the temperature? In by far the greatest number of cases salts and substances of all kinds dissolve more freely, the higher the temperature of the water ; but there are a few salts, such as calcium sulphate, which follow the opposite rule. A considerable number of salts resemble sodium sulphate in becoming more soluble up to a certain temperature, and then varying in the opposite direction. We next require to assign the amount of variation as compared with that of the temperature, assuming at first that the increase of solubility is proportional to the in- crease of temperature. Common salt is an instance of very slight variation, and potassium nitrate of very con- siderable increase with temperature. Accurate observa- tions will probably show, however, that the simple law of proportionate variation is only approximately true, and some more complicated law involving the second, third, or higher powers of the temperature may ultimately be established. All these investigations have to be carried out for each salt separately, since no distinct prin- ciples by which we may infer from one substance to another have yet been detected. There is still an in- definite field for further research open ; for the solubility k-^^*^ will probably vary with the pressure under which the medium is placed ; the presence of other salts already dissolved may, have effects vet unknown. The researches already elfected as regards the solvent power of water must be repeated with alcohol, ether, carbon bisulphide, and other media, so that unless general laws can be detected, this one phenomenon of solution can i I. I ■I it. ' II! 11 1 ' 2B0 THB PRINCIPLES OF SCIENCE. [OHAP. never be exhaustively treated. The same kind of 'questions recur as regards the solution or absorption of gases in liquids, the pressure as well as the temperature having then a most decided effect, and Professor Roscoe's re- searches on the subject present an excellent example of the successive determination of various complicated laws.* There is hardly a branch of physical science in which similar complications are not ultimately encountered. In the case of gravity, indeed, we arrive at the final law, that the force is the same for all kinds of matter, and varies only with the distance of action. But in other subjects the laws, if simple in their ultimate nature, are disguised and complicated in their apparent results. Thus the effect of heat in expanding solids, and the reverse effect of forcible extension or compression upon the tem- perature of a body, will vary from one substance to auother, will vary as the temperature is already higher or lower, and«.will probably follow a highly complex law, which in some cases gives negative or exceptional results. In crystalline substances the same researches have to be repeated in each distinct axial direction. In the sciences of pure observation, such as those of astronomy, meteorology, and terrestrial magnetism, wo meet with many interesting series of quantitative deter- minations. The so-called fixed stars, as Giordano Bruno divined, are not really fixed, and may be more truly described as vast wandering orbs, each pursuing its own path through space. We must then determine separately for each star the following questions : — 1. Does it move ? 2. In what direction ? 3. At what velocity ? 4. Is this velocity variable or uniform ? 5. If variable, according to what law ? 6. Is the direction imiform ? 7. If not, what is the form of the apparent path ? 8. Does it approach or recede ? 9. What is the form of the real path ? The successive answers to such questions in the case of certain binary stars, have aflforded a proof that the * WatU' IHctwuary of Chemistry, voL ii. p. 79a XIII.] MEASUREMENT OF PHENOMENA. 281 motions are due to a central force coinciding in law with gravity, and doubtless identical with it. In other cases the motions are usually so small that it is exceedingly difficult to distinguish them with certainty. And the time is yet far off when any general results as regards stellar motions can be established. The variation in the brightness of stars opens an un- limited field for curious observation. There is not a star in the heavens concerning which we might not have to determine : — I. Does it vary in brightness ? a. Is the brightness increasing or decreasing ? 3. Is the variation uniform ? 4. If not, acording to what law does it vary ? In a majority of cases the change will probably be found to have a periodic character, in which case several other questions will arise, such as — 5. What is the length of the period ? 6. Are there minor periods ? 7. What is the law of variation within the period ? 8. Is there any change in the amount of variation ? 9. If so, is it a secular, i.e. a continually growing change, or does it give evidence of a greater period ? Already the periodic changes of a certain number of stars have been determined with accuracy, and the lengths of the periods vary from less than three days up to intervals of time at least 250 times as great. Periods within periods have also been detected. There is, perhaps, no subject in which more complicated quantitative conditions have to be determined than ter- restrial magnetism. Since the time when the declination of the compass was first noticed, as some suppose by Columbus, we have had successive discoveries from time to time of the progressive change of declination from century to century; of the periodic character of this change; of the difference of th§ declination in various parts of the earth's surface; of the varying laws of the change of declination ; of the dip or inclination of the needle, and the corresponding laws of its periodic changes ; the horizontal and perpendicular intensities have also been the subject of exact measurement, and have been found to vary with place and time, like the directions of ' 1 t •I ; '. II 9S9 THE PRINCIPLES OF SCIENCK [chap. the needle ; daily and yearly periodic changes have also been detected, and all the elements are found to be subject to occasional storms or abnormal perturbations, in which the eleven year period, now known to be common to many planetary relations, is apparent The complete solution of these motions of the compass needle involves nothing less than a determination of its position and oscillations in every part of the world at any epoch, the like determina- tion for another epoch, and so on, time after time, until the periods of all changes are ascertained. This one sub- ject offers to men of science an almost inexhaustihle field for interesting quantitative research, in which we shall doubtless at some future time discover the operation of causes now most mysterious and unaccountable. The Methods of Acmrate Measurement. • In studying the modes hy which physicisU have ac- complished very exact measurements, we find that they are very various, but that they may perhaps be reduced under the following three classes : — I. The increase or decrease, in some determinate ratio, of the quantity to be measured, so as to bring it within the scope of our senses, and to equate it with the standard unit, or some determinate multiple or sub-multiple of this unit a. The discovery of some natural conjunction of events which will enable us to compare directly the multiples of the quantity with those of the unit, or a quantity related in a definite ratio to that unit 3. Indirect measurement, which gives us not the quan- tity itself, but some other quantity connected with it by known mathematical relations. Conditions of Acmrate Measurement. Several conditions are requisite in order that a mea- surement may be made with great accuracy, and that the results may be closely accordant when several inde- pendent measurements are made. In the first place the magnitude must be exactly defined '^ by sharp terminations, or precise marks of inconsiderable XIII.] MEASUREMENT OP PHENOMENA. 283 thickness. When a boundary is vague and graduated, like the penumbra in a lunar eclipse, it is impossible to say where the end really is, and different people will come to different results. We may sometimes overcome this difficulty to a certain extent, by observations repeated in a special manner, as we shall afterwards see ; but when possible, we should choose opportunities for measure- ment when precise definition is easy. The moment of occultation of a star by the moon can be observed with great accuracy, because the star disappears with perfect suddenness ; but there are other astronomical conjunctions, eclipses, transits, &c., which occupy a certain length of time in happening, and thus open the way to differences of opinion. It would be impossible to observe with pre- cision the movements of a body possessing no definite points of reference. The colours of the complete spectrum shade into each other so continuously that exact deter- minations of refractive indices would have been impossible, had we not the dark lines of the solar spectrum as precise points for measurement, or various kinds of homogeneous light, such as that of sodium, possessing a nearly uniform length of vibration. /p In the second place, we cannot measure accurately unless we have the means of multiplying or dividing a quantity without considerable error, so that we may correctly equate one magnitude with the multiple or sub- multiple of the other. In some cases we operate upon the quantity to be measured, and bring it into accurate coin- cidence with the actual standard, as when in photometry we vary the distance of our luminous body, until its illuminating power at a certain point is equal to that of a standard lamp. In other cases we repeat the unit until it equals the object, as in surveying land, or determining a weight by the balance. The requisites of accuracy now are :— (i) That we can repeat unit after unit of exactly equal magnitude ; (2) That these can be joined together so that the aggregate shall really be the sum of the parts. The same conditions apply to subdivision, which may be regarded as a multiplication of subordinate units. In order to measure to the thousandth of an inch, we must be able to add thousandth after thousandth without error in the magnitude of these spaces, or in their conjunction. H if ft ', il (i 'li »4 THE PRINCIPLES OF SCIENCE. [OOAP. Measuring InstrumenU, To consider the mechanical construction of scientific instruments, is no part of my purpose in this book. I wish to point out merely the general purpose of sucli instruments, and the methods adopted to cany out that purpose with great precision. In the first place we must distinguish between the instrument which effects a com- parison between two quantities, and the standard mag- nitude which often forms one of the quantities compared. The astronomer's clock, for instance, is no standard of the efflux of time; it serves but to subdivide, with approxi- mate accuracy, the interval of successive passages of a star across the meridian, which it may effect perhaps to the tenth part of a second, or agAoo part of the whole. The moving globe itself is the real standard clock, and the transit instrument the finger of the clock, while the stars are the hour, minute, and second marks, none the less accurate because they are disposed at unequal intervals. The photometer is a simple instrument, by which we com- pare the relative intensity of rays of light falling upon a given spot. The galvanometer shows the comparative intensity of electric currents passing through a wire. The calorimeter gauges the quantity of heat passing from a given object But no such instruments furnish the standard unit in terms of which our results are to be ex- pressed. In one peculiar case alone does the same instru- ment combine the unit of measurement and the means of comparison. A theodolite, mural circle, sextant, or other instrument^for the measurement of angular magnitudes has no need of an additional physical unit ; for the circle itself, or complete revolution, is the natural unit to which all greater or lesser amounts of angular magnitude are referred. The result of every measurement is to make known the purely numerical ratio existing between the magnitude to be measured, and a certain other magnitude, which should, when possible, be a fixed unit or standard magni- tude, or at least an intennediate unit of which the value can be ascertained in terms of the ultimate standard. But though a ratio is the required result, an equation is the mode in which the ratio is determined and expressed. In ZIII.] MEASUREMENT OF PHENOMENA. 285 every measurement we equate some multiple or submul- tiple of one quantity, with some multiple or submultiple of another, and equality is always the fact which we ascertain by the senses. By the eye, the ear, or the touch, we judge whether there is a discrepancy or not between two lights, two sounds, two intervals of time, two bars of metal Often indeed we substitute one sense for the other, as when the efflux of time is judged by the marks upon a moving slip of paper, so that equal intervals of time are represented by equal lengths. There is a tendency to reduce all comparisons to the comparison of space magni- tudes, but in every case one of the senses must be the ultimate judge of coincidence or nou-coincidence. Since the equation to be established may exist between any multiples or submultiples of the quantities compared, there natumlly arise several different modes of comparison adapted to different cases. Let p be the magnitude to be measured, and q that in terms of which it is to be expressed. Then we wish to find such numbers x and y, that the equation p — - q may be tyue. This equation may be presented in four forms, namely :— First Form. Second Form. Third Form. Fourth Form. ,= ., !>; = « pyqx ? = i Each of these modes of expressing the same equation cor- responds to one mode of effecting a measurement When the standard quantity is greater than that to be measured, we often adopt the first mode, and subdivide the unit until we get a magnitude equal to that measured. The angles observed in surveying, in astronomy, or in goniometry are usually smaller than a whole revolution, and the measuring circle is divided by the use of the screw and microscope, until we obtain an angle undistin- guishable from that observed. The dimensions of minute objects are determined by subdividing the inch or centi- metre, the screw micrometer being the most accurate means of subdivision. Ordinary temperatures are esti- mated by division of the standard interval between the freezing and boiling points of water, as marked on a thermometer tube. I •; I It tii! I II 7 I"- '■' 886 THE PRINCIPLES OF 80IBN0E. [OHAP. In a still greater number of cases, perhaps, we multiply the standard unit until we get a magnitude equal to that to be measured. Ordinary measurement by a foot rule, a surveyor's chain, or the excessively careful measurements of the base line of a trigonometiical survey by standard bars, are sufficient instances of this procedure. In the second case, where p - = 5, we multiply or divide a magnitude until we get what is equal to the unit, or to some magnitude easily comparable with it As a general rule the quantities which we desire to measure in physical science are too small rather than too great for easy determination, and the problem consists in multiply- ing them without introducing error. Thus the expansion of a metallic bar when heated from o* C to 100° may be multiplied by a train of levers or cog wheels. In the common thermometer the expansion of the mercury, though slight, is rendered very apparent, and easily measurable by the fineness of the tube, and many other cases might be quoted. There are some phenomena, on the contrary, which are too great or rapid to come within the easy range of our senses, and our task is then the oppo- site one of diminution. Galileo found it difficult to measure the velocity of a falling body, owing to the considerable velocity acquired in a single second. He adopted the el^ant device, therefore, of lessening the rapidity by letting the body roll down an inclined plane, which enables us to reduce the accelerating force in any required ratio. The same purpose is effected in the well-known experiments performed on Attwood's machine, and the measurement of gravity by the pendulum really depends on the same principle applied in a far more advantageous manner. Wheatstone invented a beautiful method of gal- vanometry for strong currents, which consists in drawing off from the main current a certain determinate portion, which is equated by the galvanometer to a standard current In short, he measures not the current itself but a known fraction of it In many electrical and other experiments, we wish to measure the movements of a needle or other body, which are not only very slight in themselves, but the manifes- tations of exceedingly small forces. We camiot even Zlll.] MEASCJKEMENT OP PHENOMENA. S89 approach a delicately balanced needle without disturbing it Under these circumstances the only mode of proceed- ing with accuracy, is to attach a very small mirror to the moving body, and employ a ray of light reflected from the mirror as an index of its movemenfeg. The ray may be considered quite incapable of affecting the body, and yet by allowing the ray to pass to a sufficient distance, the motions of the mirror may be increased to almost any extent A ray of light is in fact a perfectly weightless finger or index of indefinite length, with the additional advantage that the angular deviation is by the law of reflection double that of the mirror. This method was introduced by Gauss, and is now of great importance ; but in Wollaston's reflecting goniometer a ray of light had previously been employed as an index. "Lavoisier and Laplace had also used a telescope in connection with the pyrometer. It is a great advantage in some instruments that they can be readily made to manifest a phenomenon in a greater or less degree, by a very slight change in the construction. Thus either by enlarging the bulb or contracting the tube of the thermometer, we can make it give more conspicuous indications of change of temperature. The ordinary baro- meter, on the other hand, always gives the variations of pressure on one scale. The torsion balance is remark- able for the extreme delicacy which may be attained by mcreasing the length and lightness of the rod, and the length and thinness of the supporting thread. Forces so minute as the attraction of gravitation between two balls, or the magnetic and diamagnetic attraction of common Aquids and gases, may thus be made apparent, and even measui-ed. The common chemical balance, too, is capable theoretically of unlimited sensibility. The third mode of measui-ement, which may be called the Method of Repetition, is of such great importance and mterest that we must consider it in a separate section. It consists in multiplying both magnitudes to be compared until some multiple' of the first is found to coincide very nearly with some multiple of the second. If the multipli- cation can be effected to an unlimited extent, without the mtroduction of countervailing errors, the accuracy with which the required ratio can be determined is unlimited !il J ' • I THB PRINCIPLES OP SCIENOR [OBAP. and we thus account for the extraordinary precision with which intervals of time in astronomy are compared to- gether. The fourth mode of measurement, in which we equate submultiples of two magnitudes, is comparatively seldom employed, because it does not conduce to accuracy. In the photometer, perhaps, we may be said to use it ; we compare the intensity of two sources of light, by placing them both at such distances from a given surface, that the light falling on the surface is tolerable to the eye, and equally intense from each source. Since the intensity of light varies inversely as the square of the distance, the relative intensities of the luminous bodies are propor- tional to the squares of their distances. The equal in- tensity of two rays of similarly coloured light may be most accurately ascertained in the mode suggested by Arago, namely, by causing the rays to pass in opposite directions through two nearly flat lenses pressed together. There is an exact equation between the intensities of the beams when Newton's rings disappear, the ring created by one ray being exactly the complement of that created by the other. The Method of Repetition. The ratio of two quantities .can be determined with unlimited accuracy, if we can multiply both the object of measurement and the standard unit without error, and then observe what multiple of the one coincides or nearly coincides with some multiple of the other. Although per- fect coincidence can never be really attained, the error thus arising may be indefinitely reduced. For if the equation pi/ = qx be uncertain to the amount e, so that py = qx ± e^ then we have p = q -:k', and as we are supposed to be able to make y as great as we like without increasing the error e, it follows that we can make e -:- y sls small as we like, and thus approxi- mate within an inconsiderable quantity to the required ratio X -r y. This method of repetition is naturally employed when- ever quantities can be repeated, or repeat themselve* XIII.] MEASUREMENT OF PHENOMENA. 289 without error of juxtaposition, which is especially tho case with the motions of the earth and heavenly bodies. In determining the length of the sidereal day, we deter- mine the ratio between the earth's revolution round the sun, and its rotation on its own axis. We might ascertain the ratio by observing the successive passages of a star across the zenith, and comparing the interval by a good clock with that between two passages of the sun, the difference being due to the angular movement of the eaith round the sun. In such observations we should have an error of a considerable part of a second at each observation, in addition to the iiTegularities of the clock. But the revolutions of the earth repeat themselves day after day, and year after year, without the slightest in- terval between the end of one period and the beginning of another. The operation of multiplication is perfectly performed for us by nature. If, then, we can find an obser- vation of the passage of a star across the meridian a hun- dred years ago, that is of the interval of time between the passage of the sun and the star, the instrumental en-ors in measuiing this interval by a clock and telescope niay be greater than in the present day, but will be divided by about 36,524 days, and rendered excessively small It is thus that astronomei-s have been able to ascertain the ratio of the mean solar to the sidereal day to the 8th place of decimals (100273791 to i), or to the hundred millionth pait, probably the most accurate result of measui-ement in the whole range of science. The antiquity of this mode of comparison is almost as great as that of astronomy itself. Hipparchus made the first clear application of it, when he compared his own observations with those of Aristarchus, made 145 years previously, and thus ascertained the length of the year. This calculation may in fact be regarded as the earliest attempt at an exact determination of the constants of nature. The method is the main resource of astrono- mers; Tycho, for instance, detected the slow diminution of the obliquity of 'the earth's axis, by the comparison of observations at long intervals. Living astronomers use the method as much as earlier ones; but so superior m accuracy are all observations taken during the last hundred years to aU previous ones, that it is often I I I 1 i I l' M I I I ( ' S90 THE PRINCIPLES OF SCIENCE. [chap. found preferable to take a shorter interval, rather than incur the risk of greater instrumental errors in the earlier observations. It is obvious that many of the slower changes of the heavenly bodies must require the lapse of large intervals of time to render their amount perceptible. Hipparchus could not possibly have discovered the smaller inequalities of the heavenly motions, because there were no previous observations of sufficient age or exactness to exhibit them. And just as the observations of Hipparchus formed the starting-point for subsequent comparisons, so a large part of the labour of present astronomers is directed to record- ing the present state of the heavens so exactly, that future generations of astronomers may detect changes, which cannot possibly become known in the present age. The principle of repetition was very ingeniously em- ployed in an instrument first proposed by Mayer in 1767, and carried into practice in the Repeating Circle of Borda. The exact measurement of angles is indispensable, not only in astronomy but also in trigonometrical surveys, and the highest skill in the mechanical execution of the gradu- ated circle and telescope will not prevent terminal errors of considerable amount If instead of one telescope, the circle be provided with two similar telescopes, these may be alternately directed to two distant points, say the marks in a trigonometrical survey, so that the circle shall be turned through any multiple of the angle subtended by those marks, before the amount of the angular revolu- tion is read off upon the graduated circle. Theoretically speaking, all error arising from imperfect graduation might thus be indefinitely reduced, being divided by the number of repetitions. In practice, the advantage of the invention is not found to be very great, probably because a certain error is introduced at each observation in the changing and fixing of the telescopes. It is moreover inapplicable to moving objects like the heavenly bodies, so that its use is confined to important trigonometrical surveys. The pendulum is the most perfect of all instruments, chiefly because it admits of almost endless repetition. Since the force of gravity never ceases, one swing of the pendulum is no sooner ended than the other is begun, 80 that the juxtaposition of successive units is absolutely XIII.] MEASUREMENT OF PHENOMENA. 291 perfect. Provided that the oscillations be equal, one thousand oscillations will occupy exactly one thousand times as great an interval of time as one oscillation. Not only is the subdivision of time entirely dependent on this fact, but in the accurate measurement of gravity, and many other important determinations, it is of the greatest service. In the deepest mine, we could not observe the rapidity of fall of a body for more than a quarter of a minute, and the measurement of its velocity would be difficult, and subject to uncertain errors from resistance of air, &c. In the pendulum, we have a body which can be kept rising and falling for many hours, in a medium entirely under our command or if desirable in a vacuum. Moreover, the comparative force of gravity at different points, at the top and bottom of a mine for instance, can be determined with wonderful precision, by comparing the oscillations of two exactly similar pendu- lums, with the aid of electric clock signals. To ascertain the comparative times of vibration of two pendulums, it is only requisite to swing them one in front of the other, to record by a clock the moment when they coincide in swing, so that one hides the other, and then count the number of vibrations until they again come to coincidence. If one pendulum makes m vibrations and the other w, we at once have our equation pn ^ qm ; which gives the length of vibration of either pendulum in terms of the other. This method of coincidence, embody- ing the principle of repetition in perfection, was employed with wonderful skill by Sir George Airy, in his experi- ments on the Density of the Earth at the Harton Colliery, the pendulums above and below being compared with clocks, which again were compared with each other by electric signals. So exceedingly accurate was this method of observation, as earned out by Sir George Airy, that he was able to measure a total difference in the vibrations at the top and bottom of the shaft, amounting to only 2-24 seconds in the twenty-four hours, with an error of less than one hundredth part of a second, or one part in <>,o4o,ooo of the whole day.^ The principle of repetition has been elegantly applied » Philosophical Transa4iii(mt, (I856) vol 146, Part L p. 297. U 2 ■!} 29S THE PRINCIPLES OF SCIENCR ^"i h III' w 'II I i: . [chap. XIII.] MKASUKEMENT OF PHENOMENA. 293 in observing the motion of waves in water. 11 the canal in which the experiments are made be short, say twenty feet long, the waves will pass through it so rapidly that an observation of one length, as practised by Walker, will be subject to much terminal error, even when the observer is very skilful. But it is a result of the undulatory theory that a wave is unaltered, and loses no time by com- plete reflection, so that it may be allowed to tmvel back- wards and forwards in the same canal, and its motion, say through sixty lengths, or 1200 feet, may be observed with the same accuracy as in a canal 1200 feet long, with tlie advantage of greater uniformity in the condition of the canal and water.^ It is always desirable, if possible, to bring an experiment into a small compass, so that it may be well under command, and yet we may often by repetition enjoy at the same time the advantage of extensive trial. One reason of the great accuracy of weighing with a good balance is the fact, that weights placed in the same scale are naturally added together without the slightest error. There is no difficulty in the precise juxtaposition of two grams, but the juxtaposition of two metre mea- sures can only be effected with tolerable accuracy, by tlie use of microscopes and many precautions. Hence, the extreme trouble and cost attaching to the exact measure- ment of a base line for a survey, the risk of error entering at every juxtaposition of the measuring bars, and inde- fatigable attention to all the requisite precautions being necessary throughout the operation. Measurements hy Natural Coincidence, In certain cases a peculiar conjunction of circumstances enables us to dispense more or less with instrumental aids, and to obtain very exact numerical results in the simplest manner. The mere fact, for instance, that no human being has ever seen a different face of the moon from that familiar to us, conclusively proves that the period of rotation of the moon on its own axis is equal * Airy, On Tides and Wavetj Enoyclopaedia Mctropolitana, p. 345. Scott Russell, Britiih AtiocxatioH Report^ 1837, p. 432. to that of its revolution round the earth. Not only have we the repetition of these movements during 1000 or 2000 years at least, but we have observations made for us at very remote periods, free from instrumental error, no instrument being needed. We learn that the seventh satellite of Saturn is subject to a similar law, because its light undergoes a variation in each revolution, owing to the existence of some dark tract of land ; now this failure of light always occurs while it is in the same position relative to Saturn, clearly proving the equality of the axial and revolutional periods, as Huygens perceived.^ A like peculiarity in the motions of Jupiter's fourth satel- lite was similarly detected by Maraldi in 17 13. Remarkable conjunctions of the planets may sometimes allow us to compare their periods of revolution, through great intervals of time, with much accuracy. Laplace in explaining the long inequality in the motions of Jupiter and Saturn, was assisted by a conjunction of these planets, obserA^ed at Cairo, towards the close of the eleventh century. Laplace calculated that such a con- junction must have happened on the 31st of October, a.d. 1087 ; and the discordance between the distances of the planets as recorded, and as assigned by theory, was less than one-fifth part of the apparent diameter of the sun. This difference being less than the probable error of the early record, the theory was confirmed as far as facts were available.* Ancient astronomers often showed the highest inge- nuity in turning any opportunities of measurement which occurred to good account. Eratosthenes, as early as 250 B.C., happening to hear that the sun at Syene, in Upper Egypt, was visible at the summer solstice at 'the bottom of a well, proving that it was in the zenith, pro- posed to determine the dimensions of the earth, by mea- suring the length of the shadow of a rod at Alexandria on the same day of the year. He thus learnt in a rude manner the differeiice of latitude between Alexandria and Syene and finding it to be about one fiftieth part of the whole circumference, he ascertained the dimensions of the luL^^lL ^^*^^''^^*^' PP- "7,n8. Laplace's. %*<^m«, tmus- • Grant's History of Phytieal Astronomy, p. 129. THE PRINCIPLES OF SOIENCR. [chap. XIII.] MEASUREMENT OF PHENOMENA. 895 i ■ '■t ''I 1 ■ f 1 .' , ! earth within about one sixth part of the truth. The use of wells in astronomical observation appears to have been occasionally practised in comparatively recent times as by Flamsteed in 1679.' The Alexandrian astronomers employed the moon as an instrument of measurement in several sagacious modes. Wlien the moon is exactly half full, the moon, sun, and eai-th, are at the angles of a right-angled triangle. Aristarchus measured at such a time the moon's elongation from the sun, which gave him the two other angles of the triangle, and enabled him to judge of the comparative distances of tlie moon and sun from the earth. His result, though very rude, was far more accurate than any notions previously entertained, and enabled him to form some estimate of the comparative magnitudes of the bodies. Eclipses of the moon were very useful to Hipparchus in ascertaining the longtitude of the stars, which are invisible when the sun is above the horizon. For the moon when eclipsed must be 180° distant from the sun ; hence it is only requisite to measure the distance of a fixed star in longitude from the eclipsed moon to obtain with ease its angular distance from the sun. lu later times the eclipses of Jupiter have served to measure an angle; for at the middle moment of the eclipse the satellite must be in the same straight line with the planet and sun, so that we can learn from the known laws of movement of the satellite the longitude of Jupiter as seen from the sun. If at the same time we measure the elongation or apparent angular distance of Jupiter from the sun, as seen from the earth, we have all the angles of the triangle between Jupiter, the sun, and the earth, and can calculate the comparative magnitudes of the sides of the triangle by trigonometry. The transits of Venus over the sun's face are other natural events which give most accurate measurements of the sun's parallax, or apparent difference of position as seen from distant points of the earth's surface. The sun forms a kind of background on which the place of the planet is marked, and serves as a measuring instru- ment free from all the errors of construction which affect ' Baily*8 Account of Flamstud, p. lix. human instruments. The rotation of the eai-th, too, by variously affecting the apparent velocity of ingress or egress of Venus, as seen from different places, discloses the amount of the parallax. It has been suflSciently shown that by rightly choosing the moments of obser- vation, the planetary bodies may often be made to reveal their relative distance, to measure their own position, to record their own movements with a high degree of accuracy. With the improvement of astronomical instru- ments, such conjunctions become less necessary to the progress of the science, but it will always remain advan- tageous to choose those moments for observation when instrumental errors enter with the least effect. In other sciences, exact quantitative laws can occasion- ally be obtained without instrumental measurement, as when we learn the exactly equal velocity of sounds of different pitch, by observing that a peal of bells or a musical performance is heard harmoniously at any dis- tance to which the sound penetrates; this could not be the case, as Newton remarked, if one sound overtook the other. One of the most important principles of the atomic theory, was proved by implication before the use of the balance was introduced into chemistry. Wenzel observed, before 1777, that when two neutral substances decompose each other, the resulting salts are also neutral. In mixing sodium sulphate and barium nitrate, we obtain insoluble barium sulphate and neutral sodium nitrate. This result could not follow unless the nitric acid, requisite to saturate one atom of sodium, were exactly equal to that required by one atom of barium, so that an exchange could take place without leaving either acid or base in excess. An important principle of mechanics may also be established by a simple acoustical observation. When a rod or tongue of metal fixed at one end is set in vibration, the pitch of the sound may be observed to be exactly the same, whether the vibrations be small or great; hence the oscillations are isochronous, or equally rapid, independently of their magnitude. On the ground of theory, it can be shown that such a result only happens when the flexure is proportional to the deflecting force. Thus the simple observation that the pitch of 896 THE PRINCIPLES OF SCIENCE. '1 , 1 ( I s II i (•■ [cnAF. the sound of a harmonium, for inst^ince, does not chnnge with its loudness establishes an exact law of nature.* A closely similar instance is found in the proof that the intensity of light or heat rays varies inversely as the square of the distance increases. For the apparent mag- nitude certainly varies according to this law ; hence, if the intensity of light varied according to any other law, the brightness of an object would be difl'ei*ent at different distances, which is not observed to be the case. Melloni applied the same kind of reasoning, in a somewhat different form, to the radiation of heat-rays. Modes of Indirect Measurement, Some of the most conspicuously beautiful experiments in the whole range of science, have been devised for the purpose of indirectly measuring quantities, which in their extreme greatness or smallness surpass the powers of sense. All that we need to do, is to discover some other conveniently measurable phenomenon, which is re- lated in a known ratio or according to a known law, however complicated, with that to be measured. Having once obtained experimentfil data, there is no further difficulty beyond that of arithmetic or algebraic calcu- lation. Gold is reduced by the gold-beater to leaves so thin, that the most powerful microscope would not detect any measurable thickness. If we laid several hundred leaves upon each other to multiply the thickness, M'e should still have no more than ruxf^^ of an inch at the most to measure, and the errors arising in the supocposition and measurement would be considerable. But we can readily obtain an exact result through the connected amount of weight. Faraday weighed 2000 leaves of gold, each 3I inch square, and found them equal to 384 grains. From the known specific gravity of gold it was easy to calculate that the average thickness of the leaves was We must ascribe to Newton the honour of leadinu the • Jamin, Coun de Fhynqtie, vol. i. p. 152. * Faraday. Chemical Researches, p. y^\. XIII.] MEASUREMENT OF PHENOMENA 297 way in methods of minute measurement. He did not call waves of light by their right name, and did not understand their nature; yet he measured their length, though it did not exceed the 2,000,000th part of a metre or the one fifty- thousandth part of an inch. He pressed together two lenses of large but known radii. It was easy to calculate the interval between the lenses at any point, by measuring the distance from the central point of contact. Now, with homogeneous rays the successive rings of light and darkness mark the points at which the interval between the lenses is equal to one half, or any multiple of half a vibration of the light, so that the length of the vibration became known. In a similar manner many phenomena of interference of rays of light admit of the measurement of the wave lengths. Fringes of interference arise from rays of light which cross each other at a small angle, and an excessively minute dif- ference in the lengths of the waves makes a very perceptible difference in the position of the point at which two rays will interfere and produce darkness. Fizeau has recently employed Newton's rings to measure small amounts of motion. By merely counting the number of rings of sodium monochromatic light passing a certain point where two glass plates are in close proximity, he is able to ascertain with the greatest accuracy and ease the change of distance between these glasses, produced, for instance, by the expansion of a metallic bar, connected with one of the glass plates.^ Nothing excites more admiration than the mode in which scientific observers can occasionally measure quantities, which seem beyond the bounds of human observation. We know the average depth of the Pacific Ocean to be 14,190 feet, not by actual sounding, which would be impracticable in sufficient detail, but by noticing the rate of transmission of earthquake waves from the South American to the opposite coasts, the rate of movement being connected by theory with the depth of the water.' In the same way the average depth of the Atlantic Ocean is inferred to be no less than 22,157 ^eet, from the velocity ' Proceedings of the Royal SocxeiVy 30th * Herschel, Physical Oeographi/j 1 40. November, 1866. h ( 'ir I ' 898 THE PRINCIPLES OF SCIENCE. [chat. of the ordinary tidal waves. A tidal wave again gives beautiful evidence of an effect of the law of gravity, which we could never in any other way detect. Newton estimated that the moon's force in moving the ocean is only one part in 2,871400 of the whole force of gravity, so that even the pendulum, used with the utmost skill, would fail to render it apparent. Yet, the immense extent of the ocean allows the accumulation of the effect into a very palpable amount ; and from the comparative heights of the lunar and solar tides, Newton roughly estimated the comparative forces of the moon's and sun's cravitv at the earth.i ^ ^ A few years ago it might have seemed impossible that we should ever measure the velocity with which a star approaches or recedes from the earth, since the apparent position of the star is thereby unaltered. But the spec- troscope now enables us to detect and even measure such motions with considerable accuracy, by the alteration which it causes in the apparent rapidity of vibration, and conse- quently in the refrangibility of rays of Jight of definite colour. And while our estimates of the lateral move- ments of stars depend upon our very uncertain know- ledge of their distances, the spectroscope gives the motions of approach and recess irrespective of other motions except- ing that of the earth. It gives in short the motions of approach and recess of the stars relatively to the earth.* The rapidity of vibration for each musical tone, having been accurately determined by comparison with the Syren (p. 10), we can use sounds as indirect indications of rapid vibrations. It is now known that the contraction of a muscle arises from the periodical contractions of each separate fibre, and from a faint sound or susurrus which accompanies the action of a muscle, it is inferred that each contraction lasts for about one 300th part of a second Minute quantities of radiant heat are now always measured indirectly by the electricity which they produce when falJina upon a thermopile. The extreme delicacy of the method seems to be due to the power of multiplication at several points in the apparatus. The number of elements or junc- • Prindpiay bk. iii. Prop. 37, Corollaries, 3 tnd 3. Mottc'a translation, vol. ii. p. 310. * Ro6coe*8 Spectrum Analysis, ist ed. p. 296. xiii.l MEASUREMENT OF PHENOMENA. 299 tions of different metals in the thermopile can be increased so that the tension of the electric current derived from the same intensity of radiation is multiplied ; the effect of the current upon the magnetic needle can be multiplied within certain bounds, by passing the current many times round it in a coil ; the excursions of the needle can be increased by rendering it astatic and increasing the delicacy of its suspension ; lastly, the angular divei-gence can be observed, with any required accuracy, by the use of an attached mirror and distant scale viewed through a telecope (p. 287). Such is the delicacy of this method of, measuring heat, that Dr. Joule succeeded in making a thermopile which would indicate a difference of o°oooi 14 Cent.^ A striking case of indirect measurement is furnished by the revolving mirror of Wheatstone and Foucault, whereby a minute interval of time is estimated in the form of an angular deviation. Wheatstone viewed an electric spark m a mirror rotating so rapidly, that if the duration of the spark had been more than one 72,000th part of a second, the point of light would have appeared elongated to an angular extent of one-half degree. In the spark, as drawn directly from a Leyden jar, no elongation was apparent, so that the duration of the spark was immeasurably small ; but when the discharge took place through a bad conductor, the elongation of the spark denoted a sensible duration.^ In the hands of Foucault the rotating mirror gave a measure of the time occupied by light in passing through a few metres of space. Comparative Use of Measuring Instruments, In almost every case a measuring instrument serves and should serve only as a means of comparison between two or more magnitudes. As a general rule, we should not attempt to make the divisions of the measuring scale exact multiples or submultiples of the unit, but, i-egarding them as arbitrary marks, should determine their values by companson with the standard itself. The perpendicular wu^ in the field of a transit telescope, are fixed at nearly ! w l^*"^^"^' ^ratM<fij«ton« (1859), vol. cxlix. p. 94. • WatU' UxcUonary 0/ ChemiHry, yol ii. p. 393! ^ Iv ,' I 'W I It V (i i 1 4ii''' ' 300 THE PRINCIPLES OF SCIENCE. [OHAP. equal but arbitraiy distances, and those distances are afber- urards determined, as first suggested by Malvasia, by watch- ing the passage of star after star across them, and noting the intervals of time by the clock. Owing to the perfectly regular motion of the earth, these time intervals give exact determinations of the angular intervals. In the same way, the angular value of each turn of the screw micrometer attached to a telescope, can be easily and accurately ascertained. When a thermopile is used to observe radiant heat, it would be almost impossible to calculate on d priori groniidB what is the value of each division of the galvanometer circle, and still more difficult to constnict a galvanometer, so that each division should have a given value. But this is quite unnecessary, because by placing the thermopile before a body of known dimensions, at a known distance, with a known temperature and radiating power, we measure a known amount of radiant heat, and inversely measure the value of the indications of the thennopile. In a similar way Dr. Joule ascertained the actual temperature produced by the compression of bars of metal. For having inserted a small thermopile composed of a single junction of copper and iron wire, and noted the deflections of the galvanometer, he had only to dip the bars into water of different temperatures, until he produced a like deflec- tion, in order to ascertain the temperature developed by pressure.^ In some cases we are obliged to accept a very carefully constructed instrument as a standard, as in the case of a standard barometer or thermometer. But it is then best to treat all inferior instruments comparatively only, and determine the values of their scales by comparison with the assumed standard. Systematic Performance of Measurements. When a large number of accurate measurements have to be effected, it is usually desirable to make a certain number of determinations with scrupulous care, and after- wards use them as points of reference for the remaining ' Philotophieal TraiiAfteUont (1859), ^o^- "^ix. p. 119, Ac. ZIII.] MEASUREMENT OP PHENOMENA. SOI determinations. In the trigonometrical survey of a coun- try, the principal triangulation fixes the relative positions and distances of a few points with rigid accuracy. A minor triangulation refers every prominent hill or village to one of the principal points, and then the detaQs are filled in by reference to the secondary points. The survey of the heavens is effected in a like manner. The ancient astronomers compared the right ascensions of a few prin- cipal stars with the moon, and thus ascertained their posi- tions with regard to the sun; the minor stars were afterwards referred to the principal stars. Tycho followed the same method, except that he used the more slowly moving planet Venus instead of the moon. Flamsteed was in the habit of using about seven stars, favourably situated at points all round the heavens. In his early observations the distances of the other stars from these standard points were determined by the use of the quadrant.^ Even since the introduction of the transit telescope and the mural circle, tables of standard stars are formed at Greenwich, the positions being determined with all possible accuracy, so that they can be employed for purposes of reference by astronomers. In ascertaining the specific gravities of substances, all gases are referred to atmosj)heric air at a given tempera- ture and pressure ; all liquids and solids are referred to water. We require to compare the densities of water and air with great care, and the comparative densities of any two substances whatever can then be ascertained. In comparing a very great with a very small magnitude, it is usually desirable to break up the process into several steps, using intermediate terms of comparison. We should never think of measuring the distance from London to Edinburgh by laying down measuring rods, throughout the whole lengtL A base of- several miles is selected on level ground, and compared on the one hand with the standard yard, and on the other 'with the distance of London and Edinburgh, or any other two points, by trigonometrical survey. Again, it would be exceedingly difficult to com- pare the light of a star with that of the sun, which would be about thirty thousand million times greater ; but Her- * Bailj^ Accounl of FlwrnsUed, pp. 378 — 38a (I( m •I I '! ■■ 11 ■i! (I ■ I :: li SOS THE PRINCIPLES OP SCIENCE. [chap. schel ^ effects the comparison by using the full moon as an intermediate unit. Wollaston ascertained that the sun gave 801,072 times as much light as the full moon, and Herschel determined that the light of the latter exceeded that of a Centauri 27408 times, so that we find the ratio between the light of the sun and star to be that of about 22,CXX),000,000 to I. The Pendulum, By far the most perfect and beautiful of all instruments of measurement is the pendulum. Consisting merely of a heavy body suspended freely at an invariable distance from a fixed point, it is most simple in constniction ; yet all the highest problems of physical measurement depend upon its careful use. Its excessive value arises from two circum- stances. (i) The method of repetition is eminently applicable to it, as already described (p. 290). (2) Unlike other instruments, it connects together three different quantities, those of space, time, and force. In most works on natural philosophy it is shown, that when the oscillations of the pendulum are infinitely small, the square of the time occupied by an oscillation is directly proportional to the length of the pendulum, and indirectly proportional to the force affecting it, of whatever kind. The whole theory of the pendulum is contained in the formula, first given by Huygens in his Horologium Oscil- latonum. Time of oscillation = 3*14159 X A/ length of pendulu m force. The quantity 314159 is the constant ratio of the circum- ference and radius of a circle, and is of course known with accuracy. Hence, any two of the three quantities con- cerned being given, the third may be found ; or any two being maintained invariable, the third will be invariable. Thus a pendulum of invariable length suspended at the same place, where the force of gravity may be considered constant, furnishes a measure of time. The same invari- able pendulum being made to vibrate at different points of » Herschel's Aitronomyy ^817, 4th. ed. p. 553 XIII.] MEASUREMENT OF PHENOMENA. 303 the earth's surface, and the times of vibration being astro- nomically determined, the force of gravity becomes accu- rately knowa Finally, with a known force of gravity, and time of vibration ascertained by reference to the stars, the length is determinate. All astronomical observations depend upon the first manner of using the pendulum, namely, in the astrono- mical clock. In the second employment it has been almost equally indispensable. The primary principle that gravity IS equal in all matter was proved by Newton's and Gauss' pendulum experiments. The torsion pendulum of Michell, Cavendish, and Baily, depending upon exactly the same pnnciples as the ordinary pendulum, gave the density of the earth, one of the foremost natural constants. Kater and Sabine, -by pendulum observations in different parts of the earth, ascertained the variation of gravity, whence comes a determination of the earth's ellipticity. The laws of electric and magnetic attraction have also been deter- mined by the method of vibrations, which is in constant use in the measurement of the horizontal force of terres- trial magnetism. We must not confuse with the ordinary use of the pendulum its application by Neivton, to show the absence of internal friction against space,^ or to ascertain the laws of motion and elasticity.* In these cases the extent of vibration is the quantity measured, and the principles of the instrument are different Attainable Accuracy of MeasureTTient, It is a matter of some interest to compare the degrees of accuracy which can be attained in the measurement of different kinds of magnitude. Few measurements of any kind are exact to more than six significant figures,' but it IS seldom that such accuracy can be hoped for. Time is the magnitude which seems to be capable of the most exact estimation, owing to the properties of the pendulum, and the principle of repetition described in previous sections. p. 107. Prineipia, bt.ii. Sect. 6. Prop. 31. Motte's Translation, vol a !>7. 1 JS^^- ^^' '• ^^ "^- Corollary 6. Motte's Translation, vol. i p. xx Thomson and Tait's Natural Philosophy, voL I p. 333. 304 THE PRINCIPLES OF SCIENCE. [chap. xiii. K n i i 1! As regards short intervals of time, it has already been stated that Sir George Airy was able to estimate one part in 8,640,000, an exactness, as he truly remarks, " almost beyond conception." ^ The ratio between the mean solar and the sidereal day is known to be about one part in one hundred millions, or to the eighth place of decimals, (p. 289). Determinations of weight seem to come next in exact- ness, owing to the fact that repetition without error is applicable to them. An ordinary good balance should show about one part in 500,000 of the load. The finest balance employed by M. Stas, turned with one part in 825,000 of the load.' But balances have certainly been constructed to show one part in a million,^ and Ramsden is said to have constructed a balance for the Koyal Society, to indicate one part in seven millions, though this is hardly credible. Professor Clerk Maxwell takes it for granted that one part in five millions can be detected, but we ought to discriminate between what a balance can do when first constructed, and when in continuous use. Determinations of length, unless performed with extra- ordinary care, are open to much error in the junction of the measuring bars. Even in measuring the base line of a trigonometrical survey, the accuracy generally attained is only that of about one part in 60,000, or an inch in the mDe; but it is said that in four measurements of a base line carried out very recently at Cape Comorin, the greatest error was 0*077 ^^^^ ^^ ^ '^^ mile, or one part in 1 ,382,400, an almost incredible degree of accuracy. Sir J. Whitworth has shown that touch is even ap-more delicate mode of measuring lengths than sight, and by means of a splendidly executed screw, and a small cube of ii*on placed between two flat-ended iron bars, so as to be suspended when touching them, he can detect a change of dimension in a bar, amounting to no more than one-millionth of ai: inch.* * Philosophical Trtmsactioru, (1856), vol. cxlvi. pp. 330, 331. ^ First Annual Report of the Minty p. 106. 3 Jevons, ill Watts' Dictionary of Chemistry ^ vol. i. b. 483. * British Association, Glasgow, 1 856. Address of we Preside^it of ike MeehanictU Section. CHAFIER XIV. UNITS AND STANDARDS OP MEASUKEMENT. As we have seen, instruments of measurement are only means of comparison between one magnitude and another, and as a general rule we must assume some one arbitrary magnitude, in terms of which all results of measurement are to be expressed. Mere ratios be- tween any series of objects will never tell us their absolute magnitudes ; we must have at least one ratio for each, and we must have one absolute magnitude. The number of ratios n are expressible in n equations, which will contain at least n + 1 quantities, so that if we employ them to make known n magnitudes, we must have one magnitude known. Hence, whether we are measuring time, space, density, mass, weight, energy, or any other physical quantity, we must refer to some con- ci-ete standard, some actual object, which if onc^ lost and irrecoverable, all our measures lose their absolute mean- ing. This concrete standard is in all cases arbitrary in point of theory, and its selection a question of practical convenience. There are two kinds of magnitude, indeed, which do not need to be expressed in terms of arbitrary concrete units, since they pre-suppose the existence of natural standard umts. One case is that of abstract number itself, which needs no special unit, because any object which exists or IS thought of as separate fi-om other objects (p. 157) fur- nishes us with a unit, and is the only standard required. Angular magnitude is the second case in which we have a natural unit of referenci?, namely the wholo X i\ r lit 306 THE PRINCIPLES OF SCIENCE. [CHAf. revolution or perigon, as it has been called by Mr. Sande- man.^ It is a necessary result of the uniform properties of space, that all complete revolutions are equal to each other, so that we need not select any one revolution, but can always refer anew to space itself. Whether we take the whole perigon, its half, or its quarter, is really imma- terial ; Euclid took the right angle, because the Greek geo- meters had never generalised their notions of angular magnitude sufficiently to treat angles of all magnitudes, or of unlimited qxtaiUity of revolution. Euclid defines a right angle as half that made by a line with its own continuation, which is of course equal to half a revolution, but which was not treated as an angle by him. In mathematical analysis a different fraction of the perigon is taken, namely, such a fraction that the arc or portion of the circumference included within it is. equal to the radius of the circle. In this point of view angidar magnitude is an abstract ratio, namely, the ratio between the length of arc subtended and the length of the radius. The geometrical unit is then necessarily the angle corresponding to the ratio unity. This angle is equal to about 57^ 17', 44" '8, or decimally 57°-2957795 13... .* It was called by De Morgan the araial unit, but a more convenient name for common use would be radian, as suggested by Professor Everett. Though this standard angle is naturally employed in mathematical analysis, and any other unit would introduce great com- plexity, we must not look upon it as a distinct imit, since its amount is connected with that of the half perigon, by the natural constant 3*14159 . . . usutdly denoted by the letter ir. When we pass to other species of quantity, the choice of unit is found to be entirely arbitrary. Thei-e is abso- lutely no mode of defining a length, but by selecting some physical object exhibiting that length between certain obvious points — as, for instance, the extremities of a bar, or marks made upon its surface. ' Pdicoietics, or the Science of Quantity ; an Elementary Treaiiu on Algebra, and iU groundwork Arithmetic. By Archiball Saiuleinan, M.A. Cambridfje (Deighton, Bel], and Co.), 1868, p. 304. * De Morgan's Trigonometry and DouhU Algebra, p. 5. «▼.] UNITS AND STANDARDS OF MEASUREMENT. 307 Standard Unit of Time. Time is the great independent variable of all change that which itself flows on uninterruptedly, and brings the variety which we call motion and life. When we reflect upon its intimate nature. Time, like every other element of existence, proves to be an inscrutable mystery. We can only say with St. Augustin, to one who asks us what is time, "I know when you do not ask me." The mind of man will ask what can never be answered, but one result of a true and rigorous logical philosophy must be to convince us that scientific explanation can only take place V between phenomena which have something in common, and that when we get down to primary notions, like those of time and space, the mind must meet a point of mystery beyond which it cannot penetrate. A definition of time must not be looked for ; if we say with Hobbes,* that it is '* the phantasm of before and after in motion," or with Aristotle that it is " the number of motion according to former and latter," we obviously gain nothing, because the notion of time is involved in the expressions hefore and after, former and latter. Time is undoubtedly one of those primary notions which can only l)e defined physi- cally, or by observation of phenomena which proceed in time. If we have not advanced a step beyond Augustin's acute reflections on this subject,* it is curious to observe the wonderful advances which have been made in the practical measurement of its efllux. In earlier centuries the rude sun-dial or the rising of a conspicuous star gave points of reference, while the flow of water from the clepsydra, the burning of a candle, or, in the monastic ages, even' the continuous chanting of psalms, were the means of roughly subdividing periods, and marking the hours of the day and mgbt.8 The sun and stars still furnish the standard of time, but means of accurate subdivision have become requisite, and this has been furnished by the pendulum \ ^'^l^. ^^^^ «/ ^r^«- ffobbes, Edit, by Molesworth, vol. i. p. qc ^ Confesstons, bk. xi. chapters 20—28. «,^ ^* ^* ^f^ ^^^ ^^y curious particulars conceniine the measurement of time In hk AHronomy 0/ the Ancients, pp. 24i,^&a X 2 i hU nim 308 THE PRINCIPLES OF SCIENCE. [oh A p. and the chronograpli. By the pendulum we can accurately divide the day into seconds of time. By the chronograph we can subdivide the second into a hundred, a thousand, or even a million parts. "Wheats tone measured the dura- tion of an electric spark, and found it to be no more than one 115,200th part of a second, while more recently Captain Noble has been able to appreciate intervals of time not exceeding the millionth part of a second. When we come to inquire precisely what phenomenon it is tliat we thus so minutely measure, we meet insur- mountable difficulties. Newton distinguished time accord- ing as it was absolute or apparent time, in the following words : — " Absolute, true, and mathematical time, of itself and from its own nature, flows equably without regard to anything external, and by another name is called duration; relative, apparent and common time, is some sensible and external measure of duration by the means of motion."* Though we are perhaps obliged to assume the existence of a uniformly increasing quantity which we call time, yet we cannot feel or know abstract and absolute time. Duration must be made manifest to us by the recurrence of some phenomenoa The succession of our own thoughts is no doubt the first and simplest measure of time, but a very rude one, because in some persons and circumstances the thoughts evidently flow with much greater rapidity than in other persons and circumstances. In the absence of all other phenomena, the interval between one thought and another would necessarily become the unit of time, but the most cursory observations show "that there are changes in the outward world much better fitted by their constancy to measure time than the change of thoughts within us. The earth, as I have already said, is the real clock of tlie astronomer, and is practically assumed as invariable in its movements. But on what ground is it so assumed? According to the first law of motion, every body perseveres in its state of rest or of uniform motion in a right line, unless it is compelled to change that state by forces im- pressed thereon. Eotatory motion is subject to a like ^ FrineipiOf bk. i. Scholium io D^niliont. Timnfllated by Motta^ wn\. \ p. a Sec alio p. 1 1. jmr^ UNITS AND STANDARDS OF MEASUREMENT. 3()9 condition, namely, that it perseveres uniformly unless dis- turbed by extrinsic forces. Now uniform motion means motion through equal spaces in equal times, so that if we have a body entirely free from all resistance or perturba- tion, and can measure equal spaces of its path, we have a perfect measure of time. But let it be remembered that this law has never been absolutely proved by experience • for we cannot point to any body, and say that it is wholly unresisted or undisturbed ; and even if we had such a body we should need some independent standard of time to ascerUm whether its motion was really uniform As it IS m movmg bodies that we find the best standard of time we cannot use them to prove the uniformity of their own movements, which would amount to a petitio priudpii. Our experience comes to this, that when we examine and compare the movements of bodies which seem to us nearly free from disturbance, we find them giving nearly har- monious measures of time. If any one body which seems U, us to move umformly is not doing so, but is subject to hts and starts unknown to us, because we have no absolute standard of time, then all other bodies must be subject to the same arbitrary fits and starts, otherwise there would be discrepancy disclosing the irregularities. Just as in com- paring together a number of chronometers, we should soon u^^Ik .1,''''*'' ^^ *^^^' ^^^^- irregularly, as compared with the others, so m nature we detect disturbed movement by ite discrepancy from that of other bodies which we tjBUeve to be undisturbed, and which agree nearly amonc^ thenaselves. But inasmuch as the measure ot motioS involves time, and the measure of time involves motion, there must be ultimately an assumptioa We may define equal times, as times during which a moving body under the influence of no force describes equal spaces ; ^ but all we can say m support of this definition is, that it leads us into no known difficulties, and that to the best of our ex- perience one freely moving body gives the same results as any other. When we inquire where the freely moving body is. no perfectly satisfactory answer can be givea Practically the rotating globe is sufficiently accurate, and Thomson » Rankine, Fhilosophieal Magazitu, Feb. 1867, voL xxxiu p. 91. 310 THE PRINCIPLES OF SCIENCR [chap. r t •'« tf K" 4 If ? and Tait say: "Equal times are times during which the earth turns through equal angles."* No long time has passed since astronomers thought it impossible to detect any inequality in its movement. Poisson was supposed to have proved that a change in th© length of the sidereal day amounting to one ten-millionth part in 2,500 years was incompatible with an ancient eclipse recorded by the Chaldaeans, and similar calculations were made by Laplace. But it is now known that these calculations were some- what in error, and that the dissipation of enei-gy arising out of the friction of tidal waves, and the radiation of the heat into space, has slightly decrcased the rapidity of the earth's rotatory motion. The sidereal day is now longer by one part in 2,700,000, than it was in 720 B.C. Even l^efoi-e this discovery, it was known that invariability of rotation depended upon the perfect maintenance of the earth's internal heat, which is requisite in order that the earth's dimensions shall be unaltered. Now the earth being superior in temperature to empty space, must cool more or less rapidly, so that it cannot furnish an absolute measure of time. Similar objections could be raised to all other rotating bodies within our cognisance. The moon's motion round the earth, and the earth's motion round the sun, form the next best measure of time. They are subject, indeed, to disturbance from other planets, but it is believed that these perturbations must in the course of time run through their rhythmical courses, leaving the mean distances unafi'ected, an<k consequently, by the third Law of Kepler, the periodic times unchanged. But there is more reason than not to believe that the earth encounters a slight resistance in passing through space, like that which is so apparent in Encke's comet There may also be dissipation of energy in the electrical relations of the earth to the sun, possibly identical with that which is manifested in the retardation of comets.* It is probably an untrue assumption then, that the earth's orbit remains quite invariable. It is just possible that some other body may be found in the course of time to furnish a better * Treatise 011 Natural Philosophy , vol. i. p. 179. • Proceedings of the ManehetUr Philosophical ISocieiyf 28tb Not. 1871, vol xi. p. 33. xivj] UNITS AND STANDARDS OF MEASUREMENT. 311 standard of time than the earth in its annual motion. The greatly superior mass of Jupiter and its satellites, and their greater distance from the sun, may render the electrical dissipation of energy less considerable than in the case of the earth. But the choice of the best measure will always be an open one, and whatever moving body we choose may ultimately be shown to be subject to disturbing forces. The pendulum, although so admirable an instrument for subdivision of time, fails as a standard ; foi though the same pendulum affected by the same force of gravity per- forms equal vibrations in equal times, yet the slightest change in the form or weight of the pendulum, the least corrosion of any part, or the most minute displacement of the point of suspension, falsifies the results, and there enter many other difficult questions of temperature, friction, resistimce, length of vibration, &c. Thomson and Tait are of opinion * that the ultimate standard of chronometry must be founded on the physical properties of some body of more constant character than the earth ; for instance, a carefully arranged metallic spring, hermetically sealed in an exhausted glass vessel. But it is hard to see how we can be sure that the dimen- sions and elasticity of a piece of wrought metal will remain perfectly unchanged for the few millions of years contemplated by them. A nearly perfect gas, like hydrogen, is perhaps the only kind of substance in the unchanged elasticity of wliich we could have confidence. Moreover, it is difl&cult to perceive how the undulations of such a spring could be observed with the requisite accuracy. Mere recently Professor Clerk Maxwell has made the novel suggestion, discussed in a subsequent section, that undulations of light in vacuo would form the most univei-sal standard of reference, both as regards time and space. According to this system the unit of time would be the time occupied by one vibration of the par- ticular kind of light whose wave length is taken as the unit of length. * Th^ JSUmeiits of Nahtrai Philosophy, part i. p. 119. i 'I • li ii.i li ,1 II I SIS THB PRINCIPLES OF SCIENCE. [CHAF. The Unit of Space and the Bar Standard, Next in importance after the measurement of time is /y that of space. Time comes first in theory, because pheno- ^ mena, our internal thoughts for instance, may change in time without regaixi to space. As to the phenomena of outward nature, they tend more and more to resolve themselves into motions of molecules, and motion cannot be conceived or measured without reference both to time and space. Turning now to space measurement, we find it almost equally difficult to fix and define once and for ever, a unit magnitude. There are three different modes in which it has been proposed to attempt the perpetuation of a standard length. (i) By constructing an actual specimen of the standard yard or metre, in the form of a bar. (2) By assuming the globe itself to be the ultimate standard of magnitude, the practical unit being a sub- multiple of some dimension of the globe. (3) By adopting the length of the simple seconds pen- dulum, as a standard of reference. At first sight it might seem that there was no great difficulty in this matter, and that any one of these methods might serve well enough; but the more minutely we inquire into the details, the more hopeless appears to be the attempt to establish an invariable standard. We must in the first place point out a principle not^f an obvious character, namely, that the standard length must be defined hy one single object} To make two bars of exactly the sanie length, or even two bars bearing a perfectly defined ratio to each other, is beyond the power of human art If two copies of the standard metre be made and declared equally correct, future investigators will certainly discover some discrepancy between them, proving of course that they cannot both be the standard, and giving cause for dispute as to what magnitude should then be taken as correct. If one invariable bar could be constructed and main- tained as the absolute standard, no such inconvenience could arise. Each successive generation as it acquired « See Hams' i7«My ttpon Money and Coins, part ii. [1758] p. 127. iiv.] UNITS AND STANDARDS OP MEASUREMENT. 313 higher powers of measurement, would detect errors in the copies of the standard, but the standard itself would be unimpeached, and would, as it were, become by degrees more and more accurately known. Unfortunately to con- struct and preserve a metre or yard is also a task which is eitlier impossible, or what comes nearly to the same thing, cannot be shown to be possible. Passing over the practical difficulty of defining the ends of the standard length with complete accuracy, whether by dots or lines on the surface, or by the terminal points of the bar, we have no means of proving that substances remain of in- variable dimensions. Just as we cannot tell whether the rotation of the earth is uniform, except by comparing it with other moving bodies, believed to be more uniform in motion, so we cannot detect the change of length in a bar, except by comparing it with some other bar sup- posed to be invariabla But how are we to know which is the invariable bar? It is certain that many rigid and apparently invariable substances do change in di- mensions. The bulb of a thermometer certainly contracts by age, besides undergoing rapid changes of dimensions when wanned or cooled through 100** Cent. Can we be sure that even the most solid metallic bars do not slightly contract by age, or undei-go variations in their structure by change of temperature. Fizeau was induced to try whether a quartz crystal, subjected to several hundi'ed alternations of temperature, would be modified in its physical properties, and he was unable to detect any change in the coefficient of expansion.^ It does not follow, however, that, because no apparent change was discovered in a quartz crystal, newly-construct-ed bars of metal would undergo no change. The best principle, as it seems to me, upon which the ^rpetuation of a standard of length can be rested, is that, if a variation of length occurs, it will in all probability be of different amount in different substances. If then a great number of standard metres were constructed of all kmds of different metals and alloys ; hard rocks, such as granite, serpentine, slate, quartz, limestone; artificial substances, such as porcelain, glass, &c., &c., careful ' Philoiophical Maganne, (1868), 4th Series, yoL xxxvi. p. 32. H p 1> i 1 ! 314 THE PRINCIPLES OF SCIENCE. [OHAF. comparison would show from time to time the comparative variations of length of these different substances. The most variable substances would be the most divergent, and the standard would be furnished by the mean length of those which agreed most closely with each other just as uniform motion is that of those bodies which agree most closely in indicating the efflux of time. Th>e Terrestrial Standard. The second method assumes that the globe itself is a body of invariable dimensions and the founders of the me- trical system selected the ten- millionth part of the dis- tance from the equator to the pole as the definition of the metre. The first imperfection in such a method is that the earth is certainly not invariable in size; for we know that it is superior in temperature to surrounding space, and must be slowly cooling and contracting. There is much reason to believe that all earthquakes, volcanoes, mountain elevations, and changes of sea level are evidences of this contraction as asserted by Mr. Mallet.^ But such is the vast bulk of the earth and the duration of its past exis- tence, that this contraction is perhaps less rapid in propor- tion than that of any bar or other material standard which we can construct. The second and chief difficulty of this method arises from the vast size of the earth, which prevents us from making any comparison with the ultiifiate standard, ex- cept by a trigonometrical survey of a most elaborate and costly kind. The French physicists, who first proposed the method, attempted to obviate this inconvenience by carrying out the survey once for all, and then constructing a standard metre, which should be exactly the one ten millionth part of the distance from the pole to the equator. But since all measuring operations are merely approximate, it was impossible that this operation could be perfectly achieved. Accordingly, it was shown in 1838 that the supposed French metre was erroneous to the con- siderable extent of one part in 5527. It then became necessary either to alter the length of the assumed metre, * Froe4sedituu of the Boval Society, ?oth June, 1872, rq\, xx. p. 438, XI?.] UNITS AND STANDARDS OF MEASUREMENT. 316 or to abandon its supposed relation to the earth's dimen- sions. The French Government and the International Metrical Commission have for obvious reasons decided in favour of the latter course, and have thus reverted to the first method of defining the metre by a given bar. As from time to time the ratio between this assumed standard metre and the quadrant of the earth becomes more accu- rately known, we have better means of restoring that metre by reference to the globe if required. But until lost, des- troyed, or for some clear reason discredited, the bar metre and not the globe is the standard. Thomson and Tait re- mark that any of the more accurate measurements of the English trigonometrical survey might in like manner be employed to restore our standard yard, in terms of which the results are recorded. The Pendulum Standard. The third method of defining a standard length, by reference to the seconds pendulum, was first proposed by Huyghens, and was at one time adopted by the English Government. From the principle of the pendulum (p. 302) it clearly appears that if the time of oscillation and the force actuating the pendulum be the same, the length of the pendulum must be the same. We do not get rid of theoretical difficulties, for we must assume the attraction of gravity at some point of the earth's surface, say London, to be unchanged from time to time, and the sidereal day to l)e invariable, neither assumption being absolutely correct so far as we can judge. The pendulum, in short, is only an indirect means of making one physical quantity of space depend upon two other physical quan- tities of time and force; The practical difficulties are, however, of a far more serious character than the theoretical ones. The length of a pendulum is not the ordinary length of the instru- ment, which might be greatly varied without affecting the duration of a vibration, but the distance from the centre of suspension to the centre of oscillation. There are no direct means of determining this latter centre, which depend* upon the average momentum of all the particles .i'l ll ^1 f II ! 31« THE PRTNCTPLES OP SCTENCE. [cHAr i !'K I of the pendtdum as regards the centre of suspension. Huyghens discovered that the centres of suspension and oscillation are interchangeable, and Kater pointed out that if a pendulum vibrates with exactly the same rapidity when suspended from two different points, the distance between these points is the true length of the equivalent simple pendulum.^ But the practical difficulties in em- ploying Kater's reversible pendulum are considerable, and questions regarding the disturbance of the air, the force of gravity, or even the interference of electrical attractions have to be entertained. It has been shown that all the experiments made under the authority of Government for determining the ratio between the standard yard and the seconds pendulum, were vitiated by an error in the correc- tions for the resisting, adherent, or buoyant power of the air in which the pendulums were swung. Even if such corrections were rendered unnecessary by opemting in a vacuum, other difficult questions remain.* Gauss' mode of comparing the vibrations of a wire pendulum when sus- pended at two different lengths is open to equal or greater practical difficulties. Thus it is found that the pendulum standard cannot compete in accuracy and certainty with the simple bar standard, and the method would only be useful as an accessory mode of restoring the bar standard if at any time again destroyed. Unit of Density, / Before we can measure the phenomena of nature, we require a third independent unit, which shall enable us to define the quantity of matter occupying any given space. All the changes of nature, as we shall see, are probably so many manifestations of energy ; but energy requires some substratum or material machinery of molecules, in and by which it may be manifested. Observation shows that, as regards force, there may be two modes of variation of matter. As Newton says in the first definition of the Principia, ** the quantity of matter is the measure of the same, arising from its density and bulk conjunctly.** 1 Kater*8 Treatise (m Mechanics, Cabinet Cyclopadia, p. 154. * Grant's History 0/ Physical Astronomy, p. 156. XIV.] UNITS AND STANDARDS OF MEASUREMENT. SIT Thus the force required to set a body in motion varies both according to the bulk of the matter, and also accord- ing to its quality. Two cubic inches of iron of uniform quality, will require twice as much force as one cubic inch to produce a certain velocity in a given time ; but one cubic inch of gold will require more force than one cubic inch of iron. There is then some new measurable quality in matter apart from its bulk, which we may call density, and which is, strictly speaking, indicated by its capacity to resist and absorb the action of force. For the unit of density we may assume that of any substance which is uni- form in quality, and can readily be referred to from time to time. Pure water at any definite temperature, for instance that of snow melting under inappreciable pressure, fur- nishes an invariable standard of density, and by compar- ing equal bulks of various substances with a like bulk of ice-cold water, as regards the velocity produced in a unit of time by the same force, we should ascertain the densities of those substances as expressed in that of water. Practi- cally the force of gravity is used to measure density ; for a beautiful experiment with the pendulum, performed by Newton and repeated by Gauss, shows that all kinds of matter gravitate equally. Two portions of matter then which are in equilibrium in the balance, may be assumed to possess equal inertia, and their densities will therefore be inversely as their cubic dimensions. Unit of Mass, Multiplying the number of units of density of a portion of matter, by the number of units of space occupied by it, we arrive at the quantity of matter, or, as it is usually called, the unit of mass, as indicated by the inertia and gravity it possesses. - To proceed in the most simple manner, the unit of mass ought to be that of a cubic unit of matter of the standard density ; but the founders of the metrical system took as their unit of mass, the cubic centimetre of water, at the temperature of maximum density (about 4* Cent). They ceiled this unit of mass the gramme, and constructed standard specimens of the kilogram, which might be readily referred to by all who required to employ accurate weights. Unfortunately the 318 THE PRINCIPLES OP SCIENOR [OHAP. determination of the bulk of a given weight of water at a certain temperature is an operation involving many dif ficulties, and it cannot be performed in the present day with a greater exactness than that of about one part in SOOOt the results of careful observers being sometimes found to differ as much as one part in locx).* Weights, on the other hand, can be compared with each other to at least one part in a million. Hence if different specimens of the kilogram be prepared by direct weighing against water, they will not agree closely with each other ; the two principal standard kilograms agree neither with each other, nor with their definition. Accord- ing to Professor Miller the so-called Kilogramme des Archives weighs 15432-34874 grains, while the kilogram deposited at the Ministry of the Interior in Paris, as the standard for commercial purposes, weighs 1 5432*344 grains. Since a standard weight constructed of platinum, or plati- num and iridium, can be preserved free from any appreci- able alteration, and since it can be very accurately com- pared with other weights, we shaU ultimately attain the greatest exactness in our measurements of mass, by assum- ing some single kilogram as a provisional standard, leaving the determination of its actual mass in units of space and density for future investigation. This is what is practi- cally done at the present day, and thus a unit of mass takes the place of the unit of density, both in the French and English systems. The English pound is defined by a certain lump of platinum, preserved &t Westminster, and is an arbitrary mass, chosen merely that it may agree as nearly as possible with old English pounds. The gallon, the old English unit of cubic measurement, is defined by the condition that it shall contain exactly ten pounds weight of water at 62* Fahr. ; and although it is stated that it has the capacity of about 277*274 cubic inches, this ratio between the cubic and linear systems of measure- ment is not legally enacted, but left open to investigation. While the French metric system as originally designed was theoretically perfect, it does not differ practically in this point from the English system. > Olerk Maxwell's Theory of Heai^ p. 79. »ir.) UNITS AND STANDARDS OF MEASUREMENT. 319 Natural System of Standards. Quite recently Professor Clerk Maxwell has suggested that the vibrations of light and the atoms of matter might conceivably be employed as the ultimate standards of length, time, and mass. We should thus arrive at a natural system of standards, which, though possessing no present practical importance, has considerable theoretical interest " In the present state of science," he says, " the most universal standai'd of length which we could assume would be the wave-length in vacuum of a particular kind of light, emitted by some widely diffused substance such as sodium, which has well-defined lines in its spectrum. Such a standard would be independent of any changes in the dimensions of the earth, and should be adopted by those who expect their writings to be more permanent than that body." * In the same way we should get a universal standard unit of time, independent of all questions about the motion of material bodies, by taking as the unit the periodic time of vibration of that particular kind of light whose wave-length is the unit of length. It would follow that with these units of length and time the unit of velocity would coincide with the velocity of light in empty space. As regards the unit of mass. Professor Maxwell, humorously as I should think, remarks that if we expect soon to be able to determine the mass of a single molecule of some standard substance, we may wait for this deter- mination before fixing a universal standard of mass. In a* theoretical point of view there can be no reasonable doubt that vibrations of light are, as far as we can tell, the most fixed in magnitude of all phenomena. There is as usual no certainty in the matter, for the properties of the basis of light may vary to some extent in different parts of space. But no differences could ever be established in the velocity of light in different parts of the solar system, and the spectra of the stars show that the times of vibration there do not differ perceptibly from those in this part of the universe. Thus all presumption is in favour of the absolute constancy of the vibrations of light — absolute, that is, so far as regards any means of investigation we are * TrwUu on Electricity cuid Magnetiemy toL i. p. 3> i( THE PRINCIPLES OF SCIENCE. [OHAF. k It 11 ! likely to possess. Nearly the same considerations apply to the atomic weight as the standard of mass. It is im- possible to prove that all atoms of the same substance are of equal mass, and some physicists think that they differ, so that the fixity of combining proportions may bo due only to the approximate constancy of the mean of countless millions of discrepant weights. But m any case the do tection of difference is probably beyond our powers. In a theoretical point of view, then, the magnitudes suggested by Professor Maxwell seem to be the most fixed ones of wliich we have any knowledge, so that they necessarily become the natui-al units. In a practical point of view, as Professor Maxwell would be the first to point out, they are of little or no value, be- cause in the present state of science we caimot measure a vibration or weigh an atom with any approach to the accuracy which is attainable in the comparison of standard metres and kilograms. The velocity of light is not known probably within a thousandth part, and as we progress in the knowledge of light, so we shall progress in the accu- rate fixation of other standards. All that cau be said then, is that it is very desirable to determine the wave-lengths and periods of the principal lines of the solar spectrum, and the absolute atomic weights of the elements, with all attainable accuracy, in terms of our existing standards. The numbers thus obtained would^admit of the reproduc- tion of our standards in some future age of the world lo a corresponding degree of accuracy, were there need of such reference ; but so far as we can see at present, there is no considerable probability that this mode of repi*oduction would ever be the best mode. Subsidiary Units, Having once established the standard units of time, space, and density or mass, we might employ them for the expression of all quantities of such nature. But it is often convenient in particular branches of science to use mul- tiples or submultiples of the original units, for the ex- pression of quantities in a simple manner. We use the mile rather than the yard when treating of the magnitude of the globe, and the mean distance of the earth and XIV.] UNITS AND STANDARDS OP MEASUREMENT. 321 sun is not too large a unit when we have to describe the distances of the stars. On tlie other hand, when we are occupied with microscopic objects, the inch, the line or the millimetre, become the most convenient terms of expression. It is allowable for a scientific man to introduce a new unit in any branch of knowledge, provided that it assists precise expression, and is carefully brought into relation with the primary units. Thus Professor A. W. Williamson has proposed as a convenient unit of volume in chemical science, an absolute volume equal to about 11*2 litres representing the bulk of one gram of hydrogen gas at standard temperature and pressure, or the equivalent weight of any other gas, such as 16 grams of oxygen, 14 grams of nitrogen, &c. ; in short, the bulk of that quantity of any one of those gases which weighs as many grams as there are units in the number expressing its atomic weight.^ Hofmann has proposed a new unit of weight for chemists, called a crith, to be defined by the weight of one litre of hydrogen gas at 0° C. and o°76 mm., weighing about 0*0896 gram.* Both of these units must be re- garded as purely subordinate units, ultimately defined by reference to the primary units, and not involving any new assumption. Derived Units, The standard units of time, space, and mass having been once fixed, many kinds of magnitude are naturally measured by units derived from them. From the metre, the unit of linear magnitude follows in the most obvious manner the centiare or square metre, the unit of superficial magnitude, and the litre that is the cube of the tenth part of a metre, the unit of capacity or volume. Velocity of motion is ex- pressed by the ratio of the space passed over, when the motion is uniform, to the time occupied ; hence the unit of velocity is that of a body which passes over a unit of space in a unit of time. In physical science the unit of velocity might be taken as one metre per second. ^Chemistry for Stv.dents, by A. W. Williamson. Clarendon Press Senes, 2nd ed. Preface p. vi. * Introduction to Chemistry, p. 131. Y ;■ 3St THB PRINOIPLBS OP SCIENCK. [OHAf. )| t ! Momentum is measured by the mass moving, regard being paid both to the amount of matter and the velocity at which it is moving. Hence the unit of momentum will be that of a unit volume of matter of the unit density moving with the unit velocity, or in the French system, a cubic centimetre of water of the maximum density moving one metre per second. An accelerating force is measured by the ratio of the momentum generated to the time occupied, the force being supposed to act uniformly. The unit of force will therefore be that which generates a unit of momentum in a unit of time, or which causes, in the French system, one cubic centimetre of water at maximum density to acquire in one second a velocity of one metre per second. The force of gravity is the most familiar kind of force, and as, when acting unimpeded upon any substance, it produces in a second a velocity of 9-80868 . . metres per second in Paris, it follows that the absolute unit of force is about the tenth part of the force of gravity. If we employ British weights and measures, the absolute unit of force is represented by the gravity of about half an ounce, since the force of gravity of any portion of matter acting upon that matter during one second, pro- duces a final velocity of 32* 1889 feet per second or about. 32 units of velocity. AlthouglTfrom its perpetual action and approximate uniformity we find in gravity the most convenient force for reference, and thus habitually employ it to estimate quantities of matter, we must remember that it is only one of many instances of force. Strictly speaking, we should express weight in terms of force, but practically we express other forces in terms of weight We still require the unit of energy, a more com- plex notion. The momentum of a body expresses the quantity of motion which belongs or would belong to the aggregate of the particles ; but when we consider how this motion is related to the action of a force producing or removing it, we find that the effect of a force is pro- portional to the mass multiplied by the square of the velocity and it is convenient to take half this product as the expression required. But it is shown in books upon dynamics that it will be exactly the same thing if we define energy by a force acting through a space. The XIV.] UNITS AND STANDARDS OP MEASUREMENT. 323 natural unit of energy will then be that which overcomes a unit of force acting through a unit of space; when we lift one kilogram through one metre, against gravity, we therefore accomplish 9-80868 . . units of work, that is,' we lum so many units of potential energy existing in the muscles, into potential energy of gravitation. In liftin^r one pound through one foot there is in like manner a con- version of 32- 1 889 units of energy. Accordingly the unit of energy will be in the English system, that required to lift one pound through about the thirty-second part of a foot; in terms of metric units, it will be that required to lift a kilogram through about one tenth part of a metre. Every person is at liberty to measure and record quantities in terms of any unit which he likes. He may use the yard for linear measurement and the litre for cubic measurement, only there will then be a com- plicated relation between his different results. The system of derived units which we have been briefly con- sidering, is that which gives the most simple and natural relations between quantitative expressions of different kinds, and therefore conduces to ease of comprehension and saving of laborious calculation. It would evidently be a source of great convenience if scientific men could agree upon some single system of units, original and derived, in terms of which all quantities could be expressed. Statements would thus be rendered easily comparable, a large part of scientific literature would be made intelligible to all, and the saving of mental labour would be immense. It seems to be generally allowed, too, that the metric system of. weights and measures presents the best basis for the ultimate system; it is thoroughly established in Western Europe ; it is legalised in England ; it IS already commonly employed by scientific men; it is m Itself the most simple and scientific of systems. There IS every reason then why the metric system should be accepted at least in its main features. Jhrovisiorud Units, Ultimately, a« we can hardly doubt, all phenomena will be recognised as so many manifestations of energy: and, being expressed in terms of the unit of energy, wUl Y 2 3S4 THE PRINCIPLES OF SCIENCE. [CBAP. xiT.] UNITS AND STANDARDS OF MEASUREMENT. 325 )\ be referable to the primary units of space, time, and density. To effect this reduction, however, in any pwrticu- lar case, we must not only be able to compare different quantities of the phenomenon, but to trace the whole series of steps by which it is connected with the primary notions. We can readily observe that the intensity of one source of light is greater than that of another ; and, knowing that the intensity of light decreases as the square of the distance increases, we can easily determine their comparative brilliance. Hence we can express the intensity of light falling upon any surface, if we have a unit in which to make the expression. Light is un- doubtedly one form of energy, and the unit ought therefore to be the unit of energy. But at present it is quite im- possible to say how much energy there is in any particular amount of light. The question then arises, — Are we to defer the measurement of light until we can assign its relation to other forms of energy ? If we answer Yes, it is equivalent to saying that the science of light must stand still perhaps for a generation ; and not only this science but many others. The true course evidently is to select, as the provisional unit of light, some light of convenient intensity, which can be reproduced from time to time in the same intensity, and which is-defiued by physical cir- cumstances. All the phenomena of light may be experi- mentally investigated relatively to this unit, for instance that obtained after much labour by Bunsen and Eoscoe.^ In after years it will become a matter of inquiry what is the energy exerted in such unit of light ; but it may be long before the relation is exactly determined. A provisional unit, then, means one which is assumed and physically defined in a safe and reproducible manner, in order that particular quantities may be compared inUr 96 more accurately than they can yet be referred to the primary units. In reality the great majority of our measurements are expressed in terms of such provisionally independent units, and even the unit of mass, as we have seen, ought to be considered as provisional. The unit of heat ought to be simply the unit of energy, already described. But a weight can be measured to the ' rhilosophical Trantactiom (1859), vol. cxlix. p. 884, &c. one- millionth part, and temperature to less than the thousandth part of a degree Fahrenheit, and to less there- fore than the five-hundred thousandth part of the absolute temperature, whereas the mechanical equivalent of heat is probably not known to the thousandth part. Hence the need of a provisional unit of heat, which is often taken as that requisite to raise one gram of water through one degree Centigrade, that is from 0° to 1°. This quantity of heat is capable of approximate expression in terms of time, space, and mass ; for by the natural constant, determined hy Dr! Joule, and called the mechanical equivalent of heat, we know that the assumed unit of heat is equal to the energy 0^ 423*55 gram-metres, or that energy which will raise the mass of 423*55 grams through one metre against 9-8... absolute units of force. Heat, may also be expressed in terms of the quantity of ice at 0° Cent., which it is capable of converting into water under inappreciable pressure. Theory of Dimeimons, In order to understand the relations between the quan- tities dealt with in physical science, it is necessary to pay attention to the Theory of Dimensions, first clearly stated by Joseph Fourier,^ but in later years developed by several physiciste. This theory investigates the manner in which each derived unit depends upon or involves one or more of the fundamental units. The number of units in a rectan- gular area is found by multiplying together the numbers of units in the sides ; thus the unit of length enters twice into the unit of area, which is therefore said to have two dimensions with respect to length. Denoting length by L, we may say that the dimensions of area are Z x Z or Z«. It is obvious in the same way that the dimensions of volume or bulk will be L\ The number of units of mass in a body is found by nml- tiplying the number of units of volume, by those of density. Hence mass is of three dimensions as regards length, and one as regards density. Calling density D, the dimen- sions of mass are DD, As already explained, however, It is UBual to substitute an arbitrary provisional unit of • TKiom Aiudytiqiu cU la Chaleur, Paris; 1822, §§ 157- i6t. 8S6 THE PRINCIPLED OF SCIENCE. il f j [chap. mass, symbolised by M ; according to the view here taken we may say that the dimensions of M are Z*i). Introducing time, denoted by T, it is easy to see that the dimensions of velocity will be — or LT-^, because the number of units in the velocity of a body is found by dividing the units of length passed over by the units of time occupied in passing. The acceleration of a body is measured by the increase of velocity in relation to the time, that is, we must divide the units of velocity gained by the units of time occupied in gaining it ; hence its dimensions will be LT'^. Momentum is the product of mass and velocity, so that its dimensions are MLT~\ The effect of a force is measured by the acceleration produced in a unit of mass in a unit of time ; hence the dimensions of force are MLT'^. Work done is pro- portional to the force acting and to the space through which it acts ; so that it has the dimensions of force with that of length added, giving ML^T-\ It should be particularly noticed that angular mag- nitude has no dimensions at all, being measured by the ratio of the arc to the radius (p. 305). Thus we have the dimensions LL'^ or L^. This^rees with the statement previously made, that no arbitrary unit of angular mag- nitude is needed. Similarly, all pure numbers expressing ratios only, such as sines and other trigonometrical func- tions, logarithms, exponents, &c., are devoid of dimensions. They are absolute numbers necessarily expressed in terms of unity itself, and are quite unaffected by the selection of the arbitrary physical units. Angular magnitude, however, enters into other quantities, such as angular velocity, which has the dimensions —. or T-^, the units of angle being divided by the units of time occupied. The dimensions of angular acceleration are denoted by T"*. The quantities treated in the theories o,' heat and electricity are numerous and complicated as regards *hfclr dimensions. Thermal capacity has the dimensions ML~^, thermal conductivity, ML-^T~\ In Magnetism the dimensions of the strength of pole are AOL^T-\ the oip^ensions of ne^d-intensitv are M ^L~^T~\ and the XIV.] UNITS AND STANDARDS OF MEASUREMENT. 327 intensity of magnetisation has the same dimensions. In the science of electricity physicists have to deal with numerous kinds of quantity, and their dimensions are different too in the electro-static and the electro-magnetic systems. Thus electro - motive force has the dimensions M^L^T \ in the former, and M^DT^ in the latter system. Capa- city simply depends upon length in electro- statics, but upon Zf~*r» in electro-magnetics. It is worthy of par- ticular notice that electrical quantities have simple dimen- sions when expressed in terms of density instead of mass. The instances now given are sufficient to show the diffi- culty of conceiving and following out the relations of the quantities treated in physical science without a systematic method of calculating and exhibiting their dimensions. It is only in quite recent years that clear ideas about these quantities have been attained. Half a century ago pro- bably no one but Fourier could have explained what he meant by temperature or capacity for heat. The notion of measuring electricity had hardly been entertained. Besides affording us a clear view of the complex relations of physical quantities, this theory is specially useful in two ways. Firstly, it affords a test of the correctness of mathematical reasoning. According to the Principle of Homogeneity, all the quantities added together, and equated in any equation, must have the same dimensions. Hence if, on estimating the dimensions of the terms in any equa- tion, they be not homogeneous, some blunder must have been committed. It is impossible to add a force to a velo- city, or a mass to a momentum Even if the numerical values of the two members of a non-homogeneous equation were equal, this would be accidental, and any alteration in the physical units would produce inequality and disclose the fakity of the law expressed in the equation. Secondly, the theory of units enables us readily and infallibly to deduce the change in the numerical expression of any physical quantity, produced by a change in the fundamental units. It is of course obvious that in order to represent the same absolute quantity, a number must vary inversely as the magnitude of the units which are numbered. The yard expressed in feet is 3 ; taking th<i inch as the unit instead of the foot it becomes 36. Every quantity into which the dimension length enters pcaitivoly 1) I, ; M • ')l THE PRINCIPLES OF SCIENCE. [chap. must be altered in like manner. Changing the unit from the foot to the inch, numerical expressions of volume must be multiplied by 12 x 12 x 12. When a dimension enters negatively the opposite rule will hold. If for the minute we substitute the second as unit of time, then we must divide all numbers expressing angular velocities by 60, and numbers expressing angular acceleration by 60 x 60. The rule is that a numerical expression varies inversely as the magnitude of the unit as regards each whole dimension entering positively, and it varies directly as the magnitude of the unit for each whole dimension entering negatively. In the case of fractional exponents, the proper root of the ratio of change has to be taken. The study of this subject may be continued in Professor J. D. Everett's " Illustrations of the Centimetre-gramme- second System of Units," published by Taylor and Francis, 1875 ; in Professor Maxwell's " Theory of Heat ; " or Pro- fessor Fleeming Jenkin's " Text Book of Electricity." Natural Constaiits. Having acquired accurate measuring instruments, and decided upon the units in which the results shall be ex- pressed, there remains the question, What use shall be made of our powers of measurement ? Our principal object must be to discover general quantitative laws of nature ; but a very large amount of preliminary labour is employed in the accurate determination of the dimensions of existing objects, and the numerical relations between diverse forces and phenomena. Step by step every part of the material universe is surveyed and brought into known relations with other parts. Each manifestation of energy is correlated with each other kind of manifestation. Professor Tyndall has described the care with which such operations are conducted.^ " Those who are unacquainted with the details of scientific investigation, have no idea of the amount of labour expended on the determination of those numbers on which important calculations or inferences depend. They have no idea of the patience shown by a Berzelius in determining atomic weights ; by a Regnault in deter- * Tyndall's Sound, irt ed. p. 26. XIV.] UNITS AND STANDAKDS OF MEASUREMENT. 329 mining coefficients of expansion ; or by a Joule in deter- mining the mechanical equivalent of heat. There is a morality brought to bear upon such matters which, in point of severity, is probably without a parallel in any other domain of intellectual action." Eveiy new natural constant which is recorded brings many fresh inferences within our power. For if n be the number of such constants known, then J (v? — n) is the number of ratios which are within our powers of calcula- tion, and this increases with the square of n. We thus gradually piece together a map of nature, in which the lines of inference from one phenomenon to another rapidly grow in complexity, and the powers of scientific prediction are correspondingly augmented. Babbage 1 proposed the formation of a collection of the constant numbers of nature, a work which has at last been taken in hand by the Smithsonian Institution.^ It is true that a complete collection of such numbers would be almost co-extensive with scientific literature, since almost all the numbers occurring in works on chemistry, mineralogy, physics, astronomy, &c., would have to be included. Still a handy volume giving all the more important numbers and their logarithms, referred when requisite to the different units in common use, would be very useful. A small collection of constant numbers will be found at the end of Babbage's, Button's, and many other tables of logarithms, and a somewhat larger collec- tion is given in Templeton's Millvjright and Engineer'i Potket Companion, Our present object will be to classify these constant numbers roughly, according to their comparative generality and importance, under the following heads : — (i) Mathematical constants. (2) Physical constants. (3) Astronomical constants. i4) Terrestrial numbers. 5) Organic numbers. (6) Social numbers. 1 f "!:''?'* Aeeociation, Cambridge, 1833. Report, pp. 484—490. bmxlhtonxan Miscellaneous Collections, vol. xii., the Constants of mature, part. 1. Specific gravities compiled by F. W. Clarke. 8vo. washmgton, 1873. i,,id \ I ■•IN 1 1 < 1 ; If II' If : ll' ^H h P' in ' ! , 1 1 i [ 1 ' I 'i ■ I. «ao THE PRINCiPLES OF SCIBNOK I CHAP. Mathematical Constants, • At the head of the list of natural constants must come those which express the necessary relations of numbers to each other. The ordinary Multiplication Table is the most familiar and the most important of such series of constants, and is, theoretically speaking, infinite in extent. Next we must place the Arithmetical Triangle, the sig- nificance of which has already been pointed out (p. 182). Tables of logarithms also contain vast series of natural constants, arising out of the relations of pure numbers. At the base of all logarithmic theory is the mysterious natural constant commonly denoted by e, or e, being equal to the infinite series i + - + ' + — — I \- ^ 1 ^ 1.2^ 1.2.3 ^ 1.2.3.4^ ' and thus consisting of the sum of the ratios between the numbers of permutations and combinations of o, i, 2, 3, 4, &c. things. Tables of prime numbers and of the factors of composite numbers must not be forgotten. Another vast and in fact infinite series of numerical constants contains those connected with the measure- ment of angles, and embodied in trigonometrical tables, whether as natural or loganthmic sines, cosines, and tangents. It should never be forgotten that though these numbers find their chief employment in connection with trigonometry, or the measurement of the sides of a right-angled triangle, yet the numbers themselves arise out of numerical relations bearing no special relation to space. Foremost among trigonometrical constants is the well known number ir, usually employed as expressing the ratio of the circumference and the diameter of a circle ; from tr follows the value of the arcual or natural unit of angular value as expressed in ordinary degrees (p. 306). Among other mathematical constants not uncommonly used may be mentioned tables of factorials (p. 179), tables of Bemouilli's numbers, tables of the error function,^ which latter are indispensable not only in the theory of probability but also in several other branches of science. 1 J. W. L. Qlaisher, PhiUnophical Magaavu, 4th Series, yoL xlii p. 421. xiT.] UNITS AND STANDARDS OF MEASUREMENT. 381 It should be clearly underatood that the mathematical constants and tables of reference already in our possession, although very extensive, are only an infinitely small part of what might be formed. With the progress of science the tabulation of new functions will be continually demanded, and it is worthy of consideration whether public money should not be available to reward the severe, long continued, and generally thankless labour which must be gone through in calculating tables. Such labours are a benefit to the whole human race as long as it shall exist, though there are few who can appreciate the extent of this benefit. A most intere«sting and excel- lent description of many mathematical tables will be found in De Morgan's article on Tables^ in the English Cyclopmdia, Division of Arts and Sciences, vol. vii. p. 976. An almost exliaustive critical catalogue of extant tables is being published by a Committee of the British Association, two portions, drawn up chiefly by Mr. J. W. L Glaisher and Professor Cayley, having appeared in the Reports of the Association for 1873 and 1875. Physical Constants. The second class of constants contains those which refer to the actual constitution of matter. For the most part they depend upon the peculiarities of the chemical substance in question, but we may begin with those which are of the most general character. In a first sub- class we may place the velocity of light or heat undula- tions, the numbers expressing the relation between the lengths of the undulations, and the rapidity of the undulations, these numbers depending only on the pro- perties of the ethereal medium, and being probably the same in all parts of the universe. The theory of heat gives rise to several numbers of the highest importance, especially Joule's mechanical equivalent of heat, the absolute zero of temperature, the mean temperature of empty space, &c. Taking into account the diverse properties of the elements we must have tables of the atomic weights, the specific heats, the specific gravities, the refractive powers, not only of the elements, but their almost 832 THE PRINCIPLES OF SCIENCE. [chap. infinitely numerous compounds. The properties of hardness, elasticity, viscosity, expansion by heat, conducting powers for heat and electricity, must also be determined in immense detail. There are, however, certain of these numbers which stand out prominently because they serve as intermediate units or terms of comparison. Such are, for instance, the absolute coefficients of expansion of air, water and mercury, the temperature of the maximum density of water, the latent heats of water and steam, the boiling-point of water under standard pressure, the melting and boiling-points of mercury, and so forth. Astronomical Constants. The third great class consists of numbers possessing far less generality because they refer not to the properties of matter, but to the special forms and distances in which matter has been disposed in the part of the universe open to our examination. We have, first of all, to define the magnitude and form of the earth, its mean density, the constant of aberration of light expressing the relation between the earth's mean /velocity in space and the velocity of light. From the earth, as our observatory, we then proceed to lay down the mean distances of the sun, and of the planets from the same centre ; all the elements of the planetary orbits, the magnitudes, densities, masses, periods of axial rotation of the several planets are by degrees determined with growing accuracy. The same labours must be gone through for the satellites. Cata- logues of comets with the elements of their orbits, as far as ascertainable, must not be omitted. From the earth's orbit as a new base of observations, we next proceed to survey the heavens and lay down the apparent positions, magnitudes, motions, distances, periods of variation, &c. of the stars. All catalogues of stars from those of Hipparchus and Tycho, are fuU of numbers ex- pressing rudely the conformation of the visible universe. But there is obviously no limit to the labours of astrono- mers ; not only are millions of distant stars awaiting their first measurements, but those already registered require endless scrutiny as regards their movements in the three dimensions of space, their periods of revolution, their xitJ units and standards of MEASUREMENT. 338 changes of brilliance and colour. It is obvious that though astronomical numbers are conventionally called constant, they are probably in all cases subject to more or less rapid variation. Terrestrial Numbers. Our knowledge of the globe we inhabit involves many numerical determinations, which have little or no con- nection with astronomical theory. The extreme heights of the principal mountains, the mean elevations of continents, the mean or extreme depths of the oceans, the specific gravities of rocks, the temperature of mines, the host of numbers expressing the meteorological or magnetic conditions of every part of the surface, must fall into this class. Many such numbers are not to be called constant, being subject to periodic or secular changes, but they are hardly more variable in fact than some which in astronomical science are set down as constant. In many cases quantities which seem most variable may go through rhythmical changes resulting in a nearly uniform average, and it is only in the long progress of physical investigation that we can hope to discriminate successfully between those elemental num- bers which are fixed and those which vary. In the latter case the law of variation becomes the constant relation which is the object of our search. Organic Numbers. The forms and properties of brute nature having be«»Ji sufficiently defined by the previous classes of numbers, the organic world, both vegetable and animal, remains outstanding, and offers a higher series of phenomena for our investigation. All exact knowledge relating to the forms and sizes of living things, their numbsrs, the quantities of various compounds which they consume, contain, or excrete, their muscular or nervous energy, &c. must be placed apart in a class by themselves. All such numbers are doubtless more or less subject to variation, and but in a minor degree capable of exact determination. Man, so far as he is an animal, and as regards his physic/al form, must also be treated in this class. fil 384 THB PRINCIPLBS OF SCIENCB. [chap. tij. Social Numbers, Little allusion need be made in this work to the fact that man in his economic, sanitary, intellectual, aesthetic, or moral relations may become the subject of sciences, the highest and most usefiil of all sciences. Every one who is engaged in statistical inquiry must acknowledge the possibflity of natural laws governing such statistical facts. Hence we must allot a distinct place to numerical information relating to the numbers, ages, physical and sanitary condition, mortality, &c., of difiFerent peoples, in short, to vital statistics. Economic statistics, compre^ bending the quantities of commodities produced, existing, exchanged and consumed, constitute another extensive body of science. In the progress of time exact investi- gation may possibly subdue regions of phenomena which at present defy all scientific treatment That scientific method can ever exhaust the phenomena of the human mind is incredible. :.i h ' 'ii I CHAPTER XV. I I Q ANALYSIS OF QUANTITATIVE PHENOMENA. . ,' . ' 'J In the two preceding chapters we have been eng^ed in considering how a phenomenon may be accurately measured and expressed. So delicate and complex an operation is a measurement which pretends to any con siderable degree of exactness, that no small part of the skill and patience of physicists is usually spent upon this work. Much of this diflSculty arises from the fact that it is scarcely ever possible to measure a single effect at a time. The ultimate object must be to discover the mathematical equation or law connecting a quantitative cause with its quantitative effect ; this purpose usually involves, as we shall see, the varying of one condition at a time, the other conditions being maintained constant The labours of the experimentalist would be compara- tively light if he could carry out this rule of varying one circumstance at a time. He would then obtain a series of corresponding values of the variable quantities concerned, from which he might' by proper hypothetical treatment obtain the required law of connection. Bi^ in reality it is seldom possible to carry out this direction except in an approximate manner. Before then we proceed to the consideration of the actual process of quantitative induc- tion, it is necessary to review the several devices by which a complicated series of effects can be disentangled. Every phenomenon measured will usually be the sum, difference, or it may be the product or quotient, of two or more different effects, and these must be in some If I •I i\ Ml Hi i t (; .; ,1 I ' 336 THE PRINCIPLES OF SCIENCE. [CflAF. way analysed and separately measured before we possess the materials for inductive treatment. lUmtrations of the Complication of Effects, It is easy to bring forward a multitude of instances to show that a phenomenon is seldom to be observed simple and alone. A more or less elaborate process of analysis is almost always necessary. Thus if an experimentalist wishes to observe and measure the expansion of a liquid by heat, he places it in a thermometer tube and registers the rise of the column of liquid in the narrow tube. But he cannot heat the liquid without also heating the glass, 80 that the change observed is really the difference between the expansions of the liquid and the glass. More minute investigation will show the necessity perhaps of allowing for further minute effects, namely the compression of the liquid and the expansion of the bulb due to the increased pressure of the column as it becomes lengthened. In a great many cases an observed effect will be apparently at least the simple sum q£ two separate and independent effects. The hea^ evolved in the combustion of oil is partly due to the carbon and partly to the hydrogen. A measurement of the heat yielded by the two jointly, cannot inform us how much proceeds from the one and how much from the other. If by somcseparate determination we can ascertain how much the hydrogen yields, then by mere subtraction we learn what is due to the carbon; and vice vtrsd. The heat conveyed by a liquid, may be partly conveyed by true conduction, partly by convection. The light dispersed in the interior of a liquid consists both of what is reflected by floating particles and what is due to true fluorescence;^ and we must find some mode of determining one portion before we can learn the other. The apparent motion of the spots on the sun, ia the algebraic sum of the sun's axial rotation, and^of the proper motion of the spots upon the sun's surface; hence the difficulty of ascertaining by direct observations the period of the sun's rotation. We cannot obtain the weight of a portion of liquid • Stokes, Ph%l4>topliieal Trantacti&yis (1852), vo*. cxiii. p. 529. xv.] ANALYSIS OF QUANTITATIVE PHENOMENA. 337 in a chemical balance without weighing it with the containing vessel. Hence to have the real weight of the liquid operated upon in an experiment, we must make a separate weighing of the vessel, with or without the adhering film of liquid according to circumstances. This is likewise the mode in which a cart and its load are weighed together, the tare of the cart previously ascertained being deducted. The variation in the height of -the barometer is a joint effect, partly due to the real variation of the atmospheric pressure, partly to the expan- sion of the mercurial column by heat. The effects may be discriminated, if, instead of one barometer tube we have two tubes containing mercury placed closely side by side, so as to have the same temperature. If one of them be closed at the bottom so as to be unaffected by the atmo- spheric pressure, it will show the changes due to tempera- ture only, and, by subtracting these changes from those shown in the other tube, employed as a barometer, we get the real oscillations of atmospheric pressure. But this correction, as it is called, of the barometric reading, is better effected -by calculation from the readings of an ordinary thermometer. In other cases a quantitative effect will be the difference of two causes acting in opposite directions. Sir John Herschel invented an instrument like a large thermometer, which h« called the Actinometer,^ and Pouillet constructed a somewhat similar instrument called the Pyrheliometer, for ascertaining the heating power of the sun's rays. In both instruments the heat of tlie sun was absorbed by a reservoir containing water, and the rise of temperature of the water was exactly observed, either by its own expansion, or by the r^adings of a delicate thermometer immersed in it But in exposing the actinometer to the sun, we do not obtain the full effect of the heat absorbed, because the receiving surface is at the same time radiating heat into empty space. The observed increment of tem- perature is in short the difference between what is received from the sun and lost by radiation. The latter quantity is capable of ready determination ; we have only to shade the instrument from the direct rays of the sun, leaving it • Admiralty Manual of Scientific Enquiry ^ 2nd ed. p. 299. )Jii{'ii THE PRINCIPLES OF SCIENCE. [cwjO^ exposed to the sky, and we can observe how much it cools in a certain time. The total effect of the sun's rays will obviously be the apparent effect pliis the cooling effect in an equal time. By alternate exposure in sun and shade during equal intervals the desired result may be obtained with considerable accuracy.^ Two quantitative effects were beautifully distinguished in an experiment of John Canton, devised in 1761 for the purpose of demonstrating the compressibility of water. He constructed a thermometer with a large bulb full of water and a short capillary tube, the part of which above the water was freed from air. Under these circumstances the water was relieved from the pressure of the atmo- sphere, but the glass bulb in bearing that pressure was somewhat contracted. He next placed the instrument under the receiver of an air-pump, and on exhausting the air, the water sank in the tube. Having thus obtained a measure of the effect of atmospheric pressure on the bulb, he opened the top of the thermometer tube and admitted the air. The level of the water now sank still more, partly from the pressure on the buU) being now compensated, and partly from the compression of the water by the atmo- spheric pressure. It is obvious that the amount of the latter effect was approximately the difference of the two observed depressions. Not uncommonly the actual phenomenon which we wish to measure is considerably less than various disturbing effects which enter into the question. Thus the compres- sibility of mercury is considerably less than the expansion of the vessels in which it is measured under pressure, so that the attention of the experimentalist has chiefly to be concenti-ated on the change of magnitude of the vessels. Many astronomical phenomena, such as the parallax or the proper motions of the fixed stars, are far less than the errors caused by instrumental imperfections, or motions arising from precession, nutation, and aberration. We need not be surprised that astronomers have from time to time mistaken one phenomenon for another, as when Flam- steed ifjiagined that he had discovered the parallax of the Pole star.^ * Pottillet, Taylor^s Seienti/k Memoirs^ vol. iy. p. 45. ' BaUy'a AocohtU of the Hev. John Flartuteed, p. 58. XV.] ANALYSIS OF QITANTTTATIVE PHENOMENA. im 1 Methods of Eliminating Error, In any particular experiment it is the object of the ex- perimentalist to measure a single effect only, and he endeavours to obtain that effect free from interfering^ effects. If this cannot bo, as it seldom or never can really be, he makes the effect as considerable as possible compared with the other effects, which he reduces to a minimum, and treats as noxious errors. Those quantities, which are called errors in one case, may really be most important and interesting phenomena in another investiga- tion. When we speak of eliminating error we really mean disentangling the complicated phenomena of nature The physicist rightly wishes to treat one thing at a time, but as this object can seldom be rigorously carried into practice, he has to seek some mode of counteracting the irrelevant and interfering causes. ^ The general principle is that a single observation can render known only a single quantity. Hence, if several different quantitative effects are known to enter into any investigation, we must have at least as many distinct ob- servations as there are quantities to be determined. Every complete experiment wQl therefore consist in ^reneral of several operations. Guided if possible by previous know- ledge of the causes in action, we must arrange the deter- minations, so that by a simple mathematical process we may distinguish the separate quantities. There appear to be five principal methods by which we may accomplish this object ; these methods are specified below and illus- I trated in the succeeding sections. (i) The Method of Avoidance. The physicist may seek for some special mode of experiment or opportunity of obser- vation, in which the error is non-existent or inappreciable. (2) The Differential Method. He may find opportunities of observation when all interfering phenomena remain con stant, and only the subject of observation is at one time present and another time absent; the difference between two observations then gives its amount (3) The Mahod of Correction. He may endeavour to estimate the amount of the interfering effect by the best available mode, and then make a corresponding correction ui the lesults of observation. 2 2 S40 THE PRINCIPLES OP SCIENCE. [CHAf. ' I (4) The Method of Compensation. He may invent some mode of neutralising the interfering cause by balancing against it an exactly equal and opposite cause of unknown amount. (5) The Method of Reversal. He may so conduct the experiment that the interfering cause may act in opposite directions, in alternate observations, the mean result being free from interference. ^ I . Method of Avoidance of Error, Astronomers seek opportunities of observation when errors wOl be as small as possible. In spite of elaborate observations and long-continued theoretical investigation, it is not practicable to assign any satisfactory law to the refractive power of the atmosphere. Although the appa- rent change of place of a heavenly body produced by refraction may be more or less accurately calculated yet the error depends upon the temperature and pressure of the atmosphere, and, when a ray is highly inclined to the pei*pendicular, the uncertaifHy in the refraction becomes very considerable. Hence astronomers always make their observations, if possible, when the object is at the highest point of its daily course, i.e. on the meridian. In some kinds of investigation, as, for instance, in the determination of the latitude of an observatory, the astronomer is at liberty to select one or more stars out of the countless number visible. There is an evident advantage in such a case, in selecting a star which passes close to the zenith, so that it may he observed almost entirely free from atmo- spheric refraction, as was done by Hooke. Astronomers endeavour to render their clocks as accurate as possible, by removing the source of variation. The pendulum is perfectly isochronous so long as its length remains invariable, and the vibrations are exactly of equal length. They render it nearly invariable in length, that is in the distance between the centres of suspension and oscillation, by a compensatory arrangement for the change of temperature. But as tliis compensation may not be perfectly accomplished, some astronomers place their chief controlling clock in a cellar, or other apartment, where the changes of temperature may be as slight as possible. «.n-. . XV.] ANALYSIS OF QUANTITATIVE PHENOMENA. 341 At the Paris Observatory a clock has been placed in the caves beneath the building, where there is no appreciable difference between the sunimer and winter temperature. To avoid the effect of unequal oscillations Huyghens made his beautiful investigations, which resulted in the discovery that a pendulum, of which the centre of oscilla- tion moved upon a cycloidal path, would be perfectly isochronous, whatever the variation in the length of oscilla- tions. But though a pendulum may be easily rendered in some degree cycloidal by the use of a steel suspension spring, it is found that the mechanical arrangements re- quisite to produce a truly cycloidal motion introduce moi-e error than they remove. Hence astronomers seek to reduce the error to the smallest amount by maintaining their clock pendulums in uniform movement; in fact, while a clock is in good order and has the same weights, there need be little change in the length of oscillation. When a pendulum cannot be made to swing uniformly, as in experiments upon the force of gravity, it becomes re- quisite to resort to the third method, and a correction is introduced, calculated on theoretical grounds from the amount of the observed change in the length of vibration. It has been mentioned that the apparent expansion of a liquid by heat, when contained in a thermometer tube or other vessel, is the difference between the real expansion of the liquid and that of the containing vessel The effects can be accurately distinguished provided that we can learn the real expansion by heat of any one convenient liquid ; for by observing the apparent expansion of tlio same liquid in any required vessel we can by difference learn the amount of expansion of the vessel due to any given change of temperature. When we once know the change of dimensions of the vessel, we can of course deter- mine the absolute expansion of any other liquid tested in it Thus it became an all-important object in scientific research to measure with accuracy the absolute dilatation by heat of some one liquid, and mercury owing to several circumstances was by far the most suitable. Dulong and Petit devised a beautiful mode of effecting this by simply avoiding altogether the effect of the change of size of the vessel Two upright tubes full of mercury were connected by a fine tube at the bottom, and were maintained at two jiH r-; \, f I .'» il i, 342 THE PRINCIPLES OF BCIENCE. [chap. different temperatures. As mercury was free to flow from one tube to the other by the connecting tube, the two columns necessarily exerted equal pressures by the princi- ples of hydrostatics. Hence it was only necessary to mea- sure very accurately by a cathetometer the difference of level of the surfaces of the two columns of mercury, to learn the difference of length of columns of equal hydro- static pressure, which at once gives the difference of den- sity of the mercury, and the dilatation by heat. The changes of dimension in the containing tubes became a matter of entire indifference, and the length of a colunni of mercury at different temperatures was measured as easily as if it had formed a solid bar. The experiment was carried out by Regnault with many improvements of detail, and the absolute dilatation of mercury, at temperatures between o° Cent and 350°, was determined almost as accurately as was needful^ The presence of a large and unceitain amount of eiTor may render a method of experiment valueless. Foucault devised a beautiful experilnent with the pendulum for demonstrating popularly the rotation of the earth, but it could be of no use for measuring the rotation exactly. It is impossible to make the pendulum swing in a perfect plane, and the slightest lateral motion gives it an elliptic path with a progressive motion of the axis of the ellipse, wliich disguises and often entirely overpowers that due to the rotation of the earth.* Faraday's laborious experiments on the relation of gravity and electricity were much obstructed by the fact that it is impossible to move a large weight of metal without gener- ating currents of electricity, either by friction or induction. To distinguish the electricity, if any, directly due to the action of gravity from the greater quantities indirectly pi-o- duced was a problem of excessive difficulty. Baily in his experiments on the density of the earth was aware of the Bxistence of inexplicable disturbances which have since been referred with much probability to the action of electricity.* The skill and ingenuity of the experimentalist ' Jamin, Court de Physique, vol. ii. pp. 15 — 28. • Fhilosophieal Magazine, 1851, 4th Series, vol. ii. pattim. * Heam, rMlotqphiccU 7VanM0<um«, 1847, vol. cxxxni. pp. 217 xv.J ANALYSIS OF QUANTITATIVE PHENOMENA. 343 are often exhausted in trying to devise a form of apparatus in which such causes of error shall be reduced to a minimum. In some rudimentary experiments we wish merely to establish the existence of a quantitative effect without precisely measuring its amount ; if there exist causes of error of which we can neither render the amount known or inappreciable, the best way is to make them all negative so that the quantitative effects will be less than the truth rather than gieater. Grove, for instance, in proving that the magnetisation or demagnetisation of a piece of iron raises its temperature, took care to maintain the electro-magnet by which the iron was magnetised at a lower temperature than the iron, so that it would cool rather than warm the iron by radiation or conduction.^ Humfoi-d's celebrated experiment to prove that heat was' generated out of mechanical force in the boring of a cannon was subject to the difficulty that heat might be brought to the cannon by conduction from neighbouring bodies. It was an ingenious device of Davy to produce friction by a piece of clock-work resting upon a block of ice in an exhausted receiver ; as the machine rose in temperature above 32°, it was certain that no heat was received by conduction from the support.^ In many other experiments ice may be employed to prevent the access of heat by conduction, and this device, first put in practice by Murray,* is beautifully employed in Bunsen's calorimeter. To observe the true temperature of the air, though apparently so easy, is really a very difficult matter, because the thermometer is sure to be aft'ected either by the sun's rays, the radiation from neighbouring objects, or the escape of heat into space. These sources of enor are too fluctu- ating to allow of correction, so that the only accurate mode of procedure is that devised by Dr. Joule, of surrounding the thermometer with a copper cylinder ingeniously * The Corrdation of Physical Forces^ ^rd ed. p. 159. * Collected Works of Sir M, Davy, vol. ii. pp. 12—14. Elements of Chemical Philosophy, p. 94. ^ Nicholson's Journal, vol. i. p. 241 ; quoted in Treatiu on Ileat Useful Knowledge Society, p. 24. 344 THE PRINCIPLES OF SCIENCE. [CBAP. adjusted to the temperature of the air, as described by him, so that the effect of radiation shall be nullified.^ When the avoidance of error is not practicable, it will yet be desirable to reduce the absolute amount of the interfering error as much as possible before employing the succeeding methods to correct the result. As a general rule we can determine a quantity with less inaccuracy as it is smaller, so that if the eri-or itself be small the error in determining that error will be of a still lower order of magnitude. But in some cases the absolute amount of an error is of no consequence, as in the index error of a divided circle, or the difference between a chronometer and astronomical time. Even the rate at which a clock gains or loses is a matter of little importance provided it remain constant, so that a sure calculation of its amount can be made. H 2. Differential Method, When we cannot avoid the existence of error, we can often resort with success to the second mode by measuring phenomena under such circumstances that the eiTor shall remain very nearly the same in all the observations, and neutralise itself as regards the purposes in view. This mode is available whenever we want a difference between quantities and not the absolute quantity of either. The determination of the parallax of the fixed stars is exceed- ingly difficult, because the amount of parallax is far less than most of the corrections for atmospheric refraction, nutation, aberration, precession, insti-umental irregularities, &c., and can with difficulty be detected among these pheno- mena of various magnitude. But, as Galileo long ago suggested, all such difficulties would be avoided by the differential observation of stars, which, though apparently close together, are really fai separated on the line of sight. Two such stars in close apparent proximity will be sub- ject to almost exactly equal errors, so that all we need do is to observe the apparent change of place of the nearer star as referred to the more distant one. J ^1*^^^ ^IS'^)''^^^\ T^^"^ ^f ^^f P 228. Proceedings of the Manchester Phxlos(n^k%9al Society, Nov. 26. 1867, vol. vii. p' 35. XV.] ANALYSIS OF QUANTITATIVE PHENOMENA. 346 A good telescope furnished with an accurate micrometer is alone needed for the application of the method. Huyghens appears to have been the first observer who actually tried to employ the method practically, but it was not until 1835 that the improvement of telescopes and micrometers enabled Struve to detect in this way the parallax of the star a Lyrae. It is one of the many advantages of the observation of transits of Venus for the determination of the solar parallax that the refraction of the atmosphere affects in an exactly equal degree the planet and the portion of the sun's face over which it is passing. Thus the observations are strictly of a differential nature. By the process of substitutive weighing it is possible to ascertain the equality or inequality of two weights with almost perfect freedom from error. If two weights A and B be placed in the scales of the best balance vre cannot be sure that the equilibrium of the beam indicates exact equality, because the arms of the beam may be unequal or unbalanced. But if we take B out and put another weight C in, and equilibrium still exists, it is apparent that the same causes of erroneous weighing exist in both cases, supposing that the balance has not been disarranged ; B then must be exactly equal to C, since it has exactly the same effect under the same circumstances. In like manner it is a general rule that, if by any uniform mechanical process we get a copy of an object, it is unlikely that this copy will be precisely the same as the original in magnitude and form, but two copies will equally diverge from the original, and will therefore almost exactly resemble each other. Leslie's Differential Thermometer ^ was well adapted to the experiments for which it was invented. Having two equal bulbs any alteration in the temperature of the air will act equally by conduction on each and produce no change in the indications of the instrument. Only that radiant heat which is purposely thrown upon one of the bulbs will produce any effect. This thermometer in short carries out the principle of the differential method in a mechanical manner. ' Leslie, Inquiry into tht Nature of Heat, p. la \f S46 THE PRINCIPLES OF SCIEKCE. [CilAf. 3. Method of Con^ectum, Whenever the result of an experiment is affected by an interfering cause to a calculable amount, it is sufficient to add or subtract this amount We are said to correct observations when we thus eliminate what is due to extraneous causes, although of course we are only sepa- rating the correct effects of sevend agents. The variation in the height of the barometer is partly due to the change of temperature, but since the coefficient of absolute dilatation of mercury has been exactly determined, as already described (p. 341), we have only to make cal- culations of a simple character, or, what is better still, tabulate a scries of such calculations for general use, and the correction for temperature can be made with all desired accuracy. The height of the mercury in the barometer is also affected by capillary attraction, which depresses it by a constant amount depending mainly on the diameter of the tube. The requisite corrections can be estimated with accuracy sufficient for most purposes, more especially as we can check the correctness of the reading of a barometer by comparison with a standard barometer, and introduce if need be an index error including both the error in the affixing of the scale and the effect due to capillarity. But in constructing the standard barometer itself we must take greater precautions; the capillaiy depression depends somewhat upon the quality of the glass, the absence of air, and the perfect cleanliness of the mercury, so that we cannot assign the exact amount of the effect Hence a standard barometer is constructed witli a wide tube, some- times even an inch in diameter, so that the capillary effect may be rendered almost zero.^ Gay-Lussac made baro- meters in the form of a uniform siphon tube, so that the capillary forces acting at the upper and lower surfaces should balance and destroy each other ; but the method fails in practice because the lower surface, being open to the air, becomes sullied and subject to a different force of capillarity. In mechanical experiments fiiction is an interfering condition, and drains away a portion of the enei^ in- ' Jevoiji, Watts' Z>K<umaiy 0/ CluiHutry, vol i pp. 5 1 3- 5 IS- i IV.] ANALYSIS OF QUANTITATIVE PHENOMENA. 347 tended to be operated upon in a definite manner. We should of course reduce the friction in the first place to the lowest possible amount, but as it cannot be altogether pre- vented, and is not calculable with certainty from any general laws, we must determine it separately for each apparatus by suitable experiments. Thus Smeaton, in his admirable but almost forgotten researches concerning water-wheels, eliminated friction in the most simple manner by determining by trial what weight, acting by a cord and roller upon his model water-wheel, would make it turn without water as i-apidly as the water made it turn. In short, he ascertained what weight concurring with the water would exactly compensate for the friction.^ In Dr. Joule's experiments to determine the mechanical equiva- lent of heat by the condensation of air, a considerable amount of heat was produced by friction of the condensing pump, and a small portion by stirring the water employed to absorb the heat This heat of friction was measured by simply repeating the experiment in an exactly similar manner except that no condensation was effected, and ob- serving the change of tenjperature then produced.^ We may describe as test experiments any in which we perform operations not intended to give the quantity of the principal phenomenon, but some quantity which would otherwise remain as an error in the result Thus in astronomical observations almost every instrumental error may be avoided by increasing the number of observations and distributing them in such a manner as to produce in the final mean as much error in one way as in the other. But there is one source of error, first discovei*ed by Maskelyne, which cannot be thus avoided, because it affects all observations in the same direction and to the same average amount, namely the Personal Error of the observer or the inclination to record the passage of a star across the wires of the telescope a little too soon or a little too late. This personal error was first carefully described in the Edinburgh Journal of Science, voL i. p. 1 78. The difference between the jud^ent of observers at the Greenwich Observatory usually varies from j^ to J I Philotophical TramactionSy vol. IL p. 100. * Philotophical Magazitu, 3rd Series, voL xxvl p. 372. 348 THE PRINCIPLES OF SCIENCE. [chap. i V iM of a second, and remains pretty constant for the same observers.* One practised observer in Sir George Airy's pendulum experiments recorded all his time observations half a second too early on the average as compared with the chief observer.* In some observers it has amounted to seven or eight- tenths of a second.' De Morgan appears to have entertained the opinion that this source of error was essentially incapable of elimination or correction.* But it seems clear, as I suggested without knowing what had been done,^ that this personal eiTor might be determined absolutely with any desirable degree of accuracy by test experiments, consisting in making an artificial star move at a considerable distance and recording by electricity the exact moment of its passage over the wire. This method has in fact been successfully employed in Leyden, Paris, and Neuchatel.* More recently, observers were trained for the Transit of Venus Expeditions by means of a mechanical model representing the motion of Venus over the sun, this model being placed at a little distance and viewed through a telescope, so that diil'erences in the judgments of different observers would become apparent. It seems likely that tests of this nature might be employed with advantage in other cases. Newton employed the pendulum for making experi- ments on the impact of balls. Two balls were hung in contact, and one of them, being drawn aside through a measured arc, was then allowed to strike the other, the arcs of vibration giving sufficient data for calculating the distribution of energy at the moment of impact The resistance of the air was an interfering cause which he estimated very simply by causing one of the balls to make several complete vibrations without impact and then marking the reduction in the lengths of the arcs^ a proper fraction of which I'eduction was added to each of the other arcs of vibration when impact took place.^ • Oreenwich Obtervatioiu for 1866, p. xlix. • Philosophical Transactions^ 1856, p. 309. • Fenny Cyclopadioj art. Transit, voL xxv. pp. 129, 13a • Ibid. art. OhservaJtion, p. 39a * Nature, vol. i. p. 85. • Nature, voL i. p. 337. See references to the Memoirs describing ftht method. ' Frincipia, Book L Law III. Corollarj YL Scholiam. Motto't translatiou, vol. t p> 33. k IT.] ANALYSIS OP QUANTITATIVE PHENOMENA. 349 The exact definition of the standard of length is one of the most important, as it is one of the most difficult questions in physical science, and the different practice of different nations introduces needless confusion. Were all standards constructed so as to give the true length at a fixed uniform temperature, for instance the freezing- point, then any two standards could be compared without the interference of temperature by bringing them both to exactly the same fixed temperature. Unfortunately the French metre was defined by a bar of platinum at o*C, while our yard was defined by a bronze bar at 62°F. It is quite impossible, then, to make a comparison of the yard and metre without the introduction of a correction, either for the expansion of platinum or bronze, or both. Bars of metal differ too so much in their rates of ex- pansion according to their molecular condition that it is dangerous to infer from one bar to another. When we come to use instruments with great accuracy there are many minute sources of error which must be . guarded against. If a thermometer has been graduated when perpendicular, it will read somewhat differently when laid flat, as the pressure of a column of mercury is removed from the bulb. The reading may also be somewhat altered if it has recently been raised to a higher temperature than usual, if it be placed under a vacuous receiver, or if the tube be unequally heated as compared with the bulb. For these minute causes of error we may have to introduce troublesome corrections, unless we adopt the simple precaution of using the thermo- meter in circumstances of position, &c., exactly similar to those in which it was graduated. There is no end to the number of minute corrections which may ultimately be required. A large number of experiments on gases, standard weights and measures, &c., depend upon the height of the barometer ; but when experiments in dif- ferent parts of the world are compared together we ought as a further refinement to take into account the varying force of gravity, which even between London and Paris makes a difference of 'ooS inch of mercury. The measurement of quantities of heat is a matter of great difficulty, because there is no known substance impervious to heat, and the problem is therefore as l\ I 1 ( i.. I 960 THE PRINcn>LES OF 8C1ENCB. [OUAF. difficult as to measure liquids in porous vessels. To determine the latent heat of steam we must condense a certain amount of the steam in a known weight of water, and then observe the rise of temperature of the water. But while we are carrying out the experiment, part of the heat will escape by radiation and conduction from the condensing vessel or calorimeter. We may indeed reduce the loss of heat by using vessels with double sides and bright surfaces, surrounded with swans-down wool or other non-conducting materials ; and we may also avoid raising the temperature of the water much above that of the surrounding air. Yet we cannot by any such means render the loss of heat inconsiderable. Rumford ingeni- ously proposed to reduce the loss to zero by commencing the experiment when the temperature of the calorimeter is as much below that of the air as it is at the end of the experiment above it Thus the vessel will first gain and then lose by radiation and conduction, and these opposite errors will approximately balance each other. But Reg- nault has shown that the loss and gain do not proceed by exactly the same laws, so that in very accurate inves- tigations Rumford's method is not sufficient There remains the method of correction which was beautifully carried out by Regnault in his determination of the latent heat of steauL He employed two calorimeters, made in exactly the same way and alternately used to condense a certain amount of steam, so that while one was measuring the latent heat, the other calorimeter was engaged in determining the corrections to be applied, whether on account of radiation and conduction from the vessel op on account of heat reaching the vessel by means of the connecting pipes.^ 4. Method of Compensation, There are many cases in which a cause of error cannot conveniently be rendered null, and is yet beyond the reach of the third method, that of calculating the requisite correction from independent observations. The magnitude > Graham's Ch^mieal BeporU and Memoin, Cavendish Society, pp. 247, 268, &c XV.] ANALYSIS OF QtTANTITATIVE PHENOMBKA. .151 of an error may be subject to continual variations, on account of change of weather, or other fickle cirumstances beyond our controL It may either be impracticable to observe the variation of those circumstances in sufficient detail, or, if observed, the calculation of the amount of error may be subject to doubt In these cases, and only in these cases, it will be desirable to invent some artificial mode of counterpoising the variable error against an equal error subject to exactly the same variation. We cannot weigh an object with great accuracy unless we make a correction for the weight of the air displaced by the object, and add this to the apparent weight In very accurate investigations relating to standard weights, it is usual to note the barometer and thermometer at the time of making a weighing, and, from the measured bulks of the objects compared, to calculate the weight of air displaced ; the third method in fact is adopted. To make* these calculations in the frequent weighings requisite in chemical analysis would be exceedingly laborious, hence the correction is usually neglected. But when the chemist wishes to weigh gas contained in a large glass globe for the purpose of determining it^ specific gravity, the correc- tion becomes of much importance. Hence chemists avoid at once the error, and the labour of correcting it, by attaching to the opposite scale of the balance a dummy sealed glass globe of equal capacity to that containing the gas to be weighed, noting only the difference of weight when the operating globe is full and empty. The correc- tion, being the same for both globes, may be entirely neglected.^ A device of nearly the' same kind is employed in the construction of galvanometers which measure the force of an electric current by the deflection of a suspended magnetic needle. The resistance of the needle is partly due to the directive influence of the earth's magnetism, and partly to the torsion of the thread. But the former force may often be inconveniently great as well as troublesome to determine for different inclinations. Hence it is customary to connect together two equally magnetised needles, with their poles pointing in opposite directions, * Rcgnanlt's Court EUmerUaire de ChimU, 185 1, vol i p. 141. \ I -A < 85S THE PRINCIPLES OF SCIENCE. [CBAP. I Hi i / J oiie needle being within and another without the coil of wire. As regards the earth's magnetism, the needles are now astatie or indifferent, the tendency of one needle towards the pole being balanced by that of the other. An elegant instance of the elimination of a disturbing force by compensation is found in Faraday's researches upon -the magnetism of gases. To observe the magnetic attraction or repulsion of a gas seems impossible unless we enclose the gas in an envelope, probably best made of glass. But any such envelope is sure to be more or less affected by the magnet, so that it becomes difficult to distinguish between three forces wliich enter into the problem, namely, the magnetism of the gas in question, that of the envelope, and that of the surrounding atmo- spheric air. Faraday avoided all difficulties by employing two equal and similar glass tubes connected together, and •so suspended from the arm of a torsion balance that the tubes were in similar parts of the magnetic field. One tube being filled with nitrogen and the other with oxygen, it was found that the oxygen seemed to be attracted and the nitrogen repelled. The suspending thread of the balance was then turned until the force of torsion restored the tubes to their original places, where the magnetism of the tubes as well as that of the sun-ounding air, being the same and in the opposite directions upon the two tubes, could not produce any interference. The force required to restore the tubes was measured by the amount of torsion of the thread, and it indicated correctly the dif- ference between the attractive powers of oxygen and nitrogen. The oxygen was then withdrawn from one of the tubes, and a second experiment made, so as to compare a vacuum with nitrogen. No force was now required to maintain the tubes in their places, so that nitrogen was found to be, approximately speaking, indifferent to the magnet, that is, neither magnetic nor diaraagnetic, while oxygen was proved to be positively magnetic.^ It required the highest experimental skill on the part of Faraday and TyndaD, to distinguish between what is apparent and real in magnetic attraction and repulsion. Experience alone can finally decide when a com- » Tyndall*8 Faraday ^ pp. 114, 115. XV.] ANALYSIS OP QUANTITATIVE PHENOMENA. 353 pensating arrangement is conducive to accuracy. As a general rule mechanical compensation is the last resource, and in the more accurate observations it is likely to introduce more uncertainty than it removes. A multitude of instruments involving mechanical compensation have been devised, but they are usually of an unscientific character,^ because the errors compensated can be more accurately determined and allowed for. But there are exceptions to this rule, and it seems to be proved that in the delicate and tiresome operation of measuring a base line, invariable bars, compensated for expansion by heat, give ths most accurate results. This arises from the fact that it is very difiicult to determine accurately the temperature of the measuring bars under varying con- ditions of weather and manipulation.^ Again, the last refinement in the measurement of time at Greenwich Observatory depends upon mechanical compensation. Sir George Airy, observing that the standard clock increased, its losing rate 030 second for an increase of one inch in atmospheric pressure, placed a magnet moved by a baro- meter in such a position below the pendulum, as almost entirely to neutralise this cause of irregularity. The thorough remedy, however, would be to remove the cause of error altogether by placing the clock in a vacuous case. We thus see that the choice of one or other mode of eliminating an error depends entirely upon circumstances and the object in view ; but we may safely lay down the following conclusions. First of all, seek to avoid the source of error altogether if it can be conveniently done ; if not, make the experiment so that the error may be as small, but more especially as constant, as possible. If the means are at hand for determining its amount by calcula- tion from other experiments and principles of science, allow the error to exist and make a correction in the result K this cannot be accurately done or involves too much labour for the purposes in view, then throw in a counteracting error which shall as nearly as possible be of equal amount in all circumstances with that to be eliminated. There yet remains, however, one important method, that of Keversal, * Bee, for mstance, the Compensated Sympiesometer, PhUotophical Magaziney 4th Series, vol. xxxix. p. 371. * Grant, History of Phytical Asbrorwmy, pp. 146, 147. A A I 364 THE PRINCIPLES OF SCIENCE. [OHAP which will form an appropriate ti'ansition to the succeediog chapters on the Method of Mean Results and the Law of Error. iil 5. Method of Reversal, The fifth method of eliminating error is most potent and satisfactory when it can be applied, but it requires that we shall be able to reveree the apparatus and mode of procedure, so as to make the interfering cause act alternately in opposite directions. If we can get two experimental results, one of which is as much too great as the other is too small, the error is equal to half the dif- ference, and the true result is the mean of the two apparent results. It is an unavoidable defect of the chemical balance, for instance, that the points of suspen- sion of the pans cannot be fixed at exactly equal distances from the centre of suspension of the beam. Hence two weights which seem to balance each other will never be quite equal in reality. The difiference is detected by re- versing the weights, and it may be estimated by adding small weights to the deficient side to restore equilibrium, and then taking as the true weight the geometric mean of the two apparent weights of the same object If the difference is small, the arithmetic mean, that is half the sum, may be substituted for the geometric mean, from which it will not appreciably differ. This method of reversal is most extensively employed in practical astronomy. The apparent elevation of a heavenly body is observed by a telescope moving upon a divided circle, upon which the inclination of the telescope is read off". Now this reading will be erroneous if the circle and the telescope have not accurately the same centre. But if we read off* at the same time both ends of the telescope, the one reading will be about as much too small as the other is too great, and the mean will be nearly free from error. In practice the observa- tion is differently conducted, but the principle is the same ; the telescope is fixed to the circle, which moves with it, and the angle through which it moves is read off at three, six, or more points, disposed at equal intervals round the circle. The older astronomers, down even to the time oi XV.] ANALYSIS OF QUANTITATIVE PHENOMENA. 35b Flamsteed, were accustomed to use portions only of a divided circle, generally quadrants, and Eomer made a vast improvement when he introduced the complete circle. The transit circle, employed to determine the meridian passiige of heavenly bodies, is so constmcted that the telescope and the axis bearing it, in fact the whole moving part of the instrument, can be taken out of the bearing sockets and turned over, so that what was formerly the western pivot becomes the eastern one, and vice versd. It is impossible that the instrument could have been 80 perfectly constructed, mounted, and adjusted that the telescope should point exactly to the meridian, but the effect of the reversal is that it will point as much to the west in one position as it does to the east in the other, and the mean result of observations in the two positions must be free from such cause of error. The accuracy with which the inclination of the compass needle can be determined depends almost entirely on the method of reversal The dip needle consists of a bar of magnetised steel, suspended somewhat like the beam of a delicate balance on a slender axis passing through the centre of gravity of the bar, so that it is at liberty to rest in that exact degree of inclination in the magnetic meridian which the magnetism of the earth induces. The inclina- tion is read off upon a vertical divided circle, but to avoid error arising from the centring of the needle and circle, both ends are read, and the mean of the results is taken. The whole instrument is now turned carefully round through 180°, which causes the needle to assume a new position relatively to the' circle and gives two new readings, in which any error due to the wrong position of the zero of the division will be reversed. As the axis of the needle may not be exactly horizontal, it is now reversed in the same manner as the transit instrument, the end of the axis which formerly pointed east being made to point west, and a new set of four readings is taken. Finally, error may arise from the axis not passing accurately through the centre of gravity of the bar, and this error can only be detected and eliminated on chang- ing the magnetic poles of the bar by the application of a strong magnet. The error is thus made to act in opposite directions. To ensure all possible accuracy each reversal AA 2 KM THE PRINCIPLES OF SCIENCE. [chap. xw. ought to be combined with each other reversal, so that the needle will be observed in eight diPTorent positions by sixteen readings, the mean of the whole of which will give the required inclination free from all eliminable errors.* There are certain cases in which a disturbing cause can with etise be made to act in opposite directions, in alter- nate observations, so that the mean of the results will be free from disturbance. Thus in direct experiments upon the velocity of sound in passing through the air between stations two or three miles apart, the wind is a cause of error. It will be well, in the first place, to choose a time for the experiment when the air is very nearly at rest, and the disturbance slight, but if at the same moment signal sounds be made at each station and observed at the other, two sounds will be passing in opposite dii-ections tlirough the same body of air and the wind will accelerate one sound almost exactly jis it retards the other. Again, in trigonometrical surveys the apparent height of a point will be affected by atmospheric refraction and the curvature of the earth. But if in the case of two points the apparent elevation of each as seen from the other be observed, the corrections will be the same in amount, but reversed in direction, and the mean between the two apparent dif- ferences of altitude will give the true difference of level. In the next two chapters we really pursue the Method of Keversal into more complicated applications. * Quetelet, Sur la Physique du Olobe, p. 174. Janiiu, Court lU Physique, vol. i. p. 504. , t ii -i CHAPTER XVI. THE METHOD OF MEANS. All results of the measurement of continuous quantity 1 can be only approximately true. Were this assertion doubted, it could readily be proved by direct experience. If any person, using an instrument of the greatest pre- cision, makes and registers successive observations in an unbiassed manner, it ,will almost invariably be found that the results differ from each other. When we operate with sufficient care we cannot perform so simple an experiment as weighing an object in a good balance without getting discrepant numbers. Only the rough and careless experimenter will think that his observations agree, but in reality he will be found to overlook the differences. The most elaborate researches, such as those undertaken in connection with standard weights and measures, always render.it apparent that complete coinci- dence is out of the question, and that the more accurate our modes of observation are rendered, the more numerous are the sources of minute error which become apparent. We may look upon the existence of error in all measure- ^ ments as the normal state of things. It is absolutely ' impossible to eliminate separately the multitude of small disturbing influences, except by balancing them off against each other. Even in drawing a mean it is to be expected that we shall come near the truth rather than exactly to it In the measurement of continuous quantity, absolute coincidence, if it seems to occur, must be only apparent, and is no indication of precision. It is one of the most embarrassing things we can meet when experimental I ii a i 358 THE PRINCIPLES OF SCIENCE. [chap. As restdts agree too closely. Such coincidences should raise ! our suspicion that the apparatus in use is in some way restricted in its operation, so as not really to give the true result at all, or that the actual results have not heen faith- fully recorded by the assistant in charge of the apparatus. If then we cannot get twice over exactly the same result, the question arises, How can we ever attain the truth or select the result which may be supposed to approach most nearly to it ? The quantity of a certain phenomenon is expressed in several numbers which differ from each other ; no more tlian one of them at the most can be true, and it is more probable that they are all false. It may be suggested, perhaps, that the observer should select the one observation which he judged to be the best made, and there will often doubtless be a feeling that one or more results were siitisfactory, and the others less trustworthy. This seems to have been the course adopted by the early astronomers. Flamsteed, when he had made several observations of a star, probably chose in an arbitrary manner that which 'seemed to him nearest to the truth.^ When Horrocks selected for his estimate of the sun's semi-diameter a mean between the results of Kepler and Tycho, he professed not to do it from any regard to the idle adage, "Medio tutissimus ibis," but because he thought it from his own observations to be correct* But this method will not apply at all when the obsei-ver has made a number of measurements which are equally good in his opinion, and it is quite apparent that in using an instrument or apparatus of considerable complication the observer will not necessarily be able to judge whether slight causes have affected its operation or not. In this question, as indeed throughout inductive logic, we deal only with probabilities. There is no infallible mode of amving at the absolute truth, which lies beyond the reach of human intellect, and can only be the distant object of our long-continued and painful approximations. Nevertheless there is a mode pointed out alike by common sense and the highest mathematical reasoning, which is ' Baily^s Account of Flamstscdy p. 376. ' The TrantU of Vmut acrou the Sun^ by Horrock^ LondoD, 1859. p. 146. XVI.] THE BIETHUD OF MEANS. 359 more likely than any other, as a general rule, to bring us near the truth. The apiarov ficTpov, or the aurea mediocritas, was highly esteemed in the ancient philosophy of Greece and Rome ; but it is not probable that any of the ancients should have been able clearly to analyse and express the reasons why they advocated the Toean as the safest course. But in the last two centuries this apparently simple question of the mean has been found to afford a field for the exercise of the utmost mathematical skill. Roger Cotes, the editor of the Principia, appears to have had some insight into the value of the mean ; but profound mathematicians such as De Moivre, Daniel Bernoulli, Laplace, Lagrange, Gauss, Quetelet, De Morgan, Airy, Leslie Ellis, Boole, Glaisher, and others, have hardly ex- hausted the subject Several uses of the Mean Result. The elimination of errors of unknown sources, is almost always accomplished by the simple arithmetical process of taking the mean, or, as it is often called, the average of several discrepant numbers. To take an average is to add the several quantities together, and divide by the number of quantities thus added, which gives a quotient lying among, or in the middle of, the several quantities. Before however inquiring fully into the grounds of this procedure, it is essential to observe that this one arith- metical process is really applied in at least three different cases, for different purposes, and upon different principles, and we must take great care not to confuse one applica- tion of the process with another. A m^n result, then, may have any one of the following significations. (1) It may give a merely representative number, expressing the general magnitude of a series of quantities, and serving as a convenient mode of comparing them with other series of quantities. Such a number is properly called Thefictitwus mean or The average result. (2) It may give a result approximately free from disturbing quantities, which are known to affect some results in one dii-ection, and other results equally in the opposite direction. We may say that in this case we get a Precise mean result. 300 THE PRINCIPLES OP SCIBI CB. [OHAP. 1VI.J THE METHOD OF MEANS. 361 it (3) It may give a result more or less free from imknown and uncertain errors; this we may call the Probable mean result. Of these three uses of the mean the first is entirely dif- ferent in nature from the two last, since it does not yield an approximation to any natural quantity, but furnishes us with an arithmetic result comparing the aggregate of certain quantities with their number. The third use of the mean rests entirely upon the theory of probability, and will be more fully considered in a later part of this chapter. The second use is closely connected, or even identical with, the Method of Keversal already described, but it will be desirable to enter somewhat fully into all the three employments of the same arithmetical process. 7.%e Mean and the Average. Much confusion exists in the popular, or even the scientific employment of the terms mean and average, and they are commonly taken as synonymous. It is necessar}' to ascertain caiefully what significations we ought to attach to them. The English word m^an is equivalent to medium, being derived, perhaps through the French moijen, from the Latin mcdius, which again is undoubtedly kindred with the Greek fMcaofi. Etymologists l>elieve, too, that this Greek word is connected with the preposition fiera, the German miite, and the true English viid or middle ; so that after all the m^^an is a technical term identical in its root with the more popular equivalent middle. If we inquire what is the mean in a mathematical point of view, the true answer is that there are several or many kinds of means. The old arithmeticians recognised ten kinds, which are stated by Boethius, and an eleventh was added by Jordanus.* The arithmetic m^ean is the one by far the most commonly denoted by the term, and that which we may understand it to signify in the absence of any qualification. It is the sum of a series of quantities divided by their number, and may be represented by the formula i (a + b). ' De Morgan, Sapplement to the Penny Oyelopadia, art. Old AppeUatimu of Nwnberg. But there is also the geometric m^an, which is the square root of the product, V<» X b, or that quantity the loga- rithm of which is the arithmetic mean of the logarithms of the quantities. There is also the harmonic mean, which is the reciprocal of the arithmetic mean of the reciprocals of the quantities. Thus if a and b be the quantities, as before, their reciprocals are - and r, the mean of which is ^ ^- 4. i), and the reciprocal again is — p-,, which is the harmonic mean. Other kinds of means might no doubt be invented for particular purposes, and we might apply the term, as De Morgan pointed out,^ to any quantity a function of which is equal to a function of two or more other quantities, and is such that the interchange of these latter quantities among them- selves will make no alteration in the value of the function. SymbolicaUy, if 4)(y,y,y ) = (pt^, «,, ajg . . . .), then y is a kind of mean of the quantities, Xi, x^, &c. The geometric mean is necessarily adopted in certain cases. When we estimate the work done against a force which varies inversely as the square of the distance from a fixed point, the mean force is the geometric mean between the forces at the beginning and end of the path. When in an imperfect balance, we reverse the weights to eliminate error, the true weight will be the geometric mean of the two apparent weights. In almost all the calculations of statistics and commerce the geometric mean ought, strictly speaking, to be used. If a commodity rises in price 100 per cent, and another remains unaltered, the mean rise of a price is not 50 per cent, because the ratio 150 : 200 is n ot the same as 100 : 150. The mean ratio is as unity to s/roo X 200 or I to 1*41. The difference between the three kinds of means in such a case * is very considerable ; while the rise of price estimated by the Arithmetic mean would be 50 per cent, it would be only 41 and 33 per cent, respectively according to the Geometric and Harmonic means. ' Penny Cydopcedia, art Mean. * Jevons, Journal of the Statistical Society f June 1865, ^^l' ^cxviii p. 296. I , THE PRINCIPLES OF SCIENCE. [OIIAP In all calculations concerning the average rate of progress of a community, or any of its operations, the geometric mean should be employed. For if a quantity increases loo per cent, in loo years, it would not on the average increase lo per cent, in each ten years, as the lo per cent, would at the end of each decade be calculated upon larger and larger quantities, and give at the end of too years much more than loo per cent., in fact as much as 159 per ^ent. The true mean rate in each decade would be *J/2~ or about 107, that is, the increase would be about 7 per cent, in each ten years. But when the quantities differ very little, the arithmetic and geometric means are approximately the same. Thus the arithmetic mean of rcxxD and i 001 is i 0005, and the geometric mean is about I 0004998, the difference being of an oitler in- appreciable in almost all scientific and practical matters. Even in the comparison of standard weights by Gauss' method of reversal, the arithmetic mean may usually be substituted for the geometric mean which is the true result Regarding the mean in the absence of express qualifica- tion to the contrary as the common arithmetic mean, we must still distinguish between its two uses where it gives with more or less accuracy and probability a really existing quantity, and where it acts as a mere representative of other quantities. If I make many experiments to determine the atomic weight of an element, there is a certain immber which I wish to approximate to, and the mean of my separate results will, in the absence of any reasons to the contrary, be the most probable approximate result When we determine the mean density of the earth, it is not because any part of the earth is of that exact density ; there may be no part exactly corresponding to the mean density, and as the crust of tlie earth has only about half the mean density, the internal matter of the globe must of course be above the mean. Even the density of a homogeneous substance like carbon or gold must be regarded as a mean between the real density of its atoms, and the zero density of the interven- nig vacuous space. The very different signification of the word " mean " in these two uses was fully explained by Quetelet,^ and the * Liters <m iht Tluory of ProbabilUia, transl. by Downes, Part iL «T1.] THE METHOD OF MEANS. 363 importance of the distinction was pointed out by Sir John Herschel in reviewing his work.^ It is much to be desired that scientific men would mark the difference by using the word mean only in the former sense when it denotes ap- proximation to a definite existing quantity ; and average, when the mean is only a fictitious quantity, used for con- venience of thought and expression. The etymology of this word " average " is somewhat obscure ; but according to De Morgan * it comes from aver-ia, " havings or pos- sessions," especially applied to farm stock. By the acci- dents of language averagium came to mean the labour of farm horses to which the lord was entitled, and it prob- ably acquired in this manner the notion of distributing a whole into parts, a sense in which it was early applied to maritime averages or contributions of the other owners of cargo to those whose goods have been thrown overboard or used for the safety of the vessel. On ilve Average or FicHtums Mean. Although the average when employed in its proper sense of a fictitious mean, represents no really existing quantity, it is yet of the highest scientific importance, as enabling us to conceive in a single result a multitude of details. It enables us to make a hypothetical simplifica- tion of a problem,and avoid complexity without committing error. The weight of a body is the sum of the weights of infinitely small particles, each acting at a different place, so that a mechanical problem resolves itself, strictly speak- ing, into an infinite number of distinct problems. We owe to Archimedes the first introduction of the beautiful idea that one point may be discovered in a gravitating body such that the weight of all the particles may be re- garded as concentrated in that point, and yet the behaviour of the whole body will be exactly represented by the behaviour of this heavy point This Centre of Gravity may be within the body, as in the case of a sphere, or it may be in empty space, as in the case of a ring. Any two bodies, whether comiected or separate, may be conceived 1 HerscheVs Essaytf &c ppi 404, 405. * On the Theory of Errors of Observationtf Camhridge FhUotophical Transactions, yoL x. Part ii 416. < u fj I) i 364 THE PRINCIPLES OP SCIENCE. [CHAF. , ^1 V t ' as having a centre of gravity, that of the sun and earth lying within the sun and only 267 miles from its centre. Although we most commonly use the notion of a centre or average point with regard to gravity, the same notion is applicable to other cases. Terrestrial gravity is a case of approximately parallel forces, and the centre of gravity is but a special case of the more general Centre of Parallel Forces. Wherever a number of forces of whatever amount act in parallel lines, it is possible to discover a point at which the algebraic sum of the forces may be imagined to act with exactly the same effect Water in a cistern presses against the side with a pressure varying according to the depth, but always in a direction perpendicular to the side. We may then conceive the whole pressure as exerted on one point, which will be one-third from the bottom of the cistern, and may be called the Centre of Pressure. The Centre of Oscillation of a pendulum, dis- covered by Huyghens, is that point at which the whole weight of the pendulum may be considered as concentrated, without altering the time of oscillation (p. 315). When one body strikes another the Centre of Percussion is that point in the striking body at which all its mass might be concentrated without altering the effect of the stroke. In position the Centre of Percussion does not differ from the Centre of Oscillation. Mathematicians have also described the Centre of Gyration, the Centre of Convei-sion, the Centre of Friction, &c. We ought carefully to distinguish between those cases in which an invarialle centre can be assigned, and those in which it cannot. In perfect strictness, there is no such thing as a true invariable centre of gravity. As a general rule a body is capable of possessing an invariable centre only for perfectly parallel forces, and gravity never does act in absolutely parallel lines. Thus, as usual, we find that our conceptions are only hypothetically correct, and only approximately applicable to real circumstances. There are indeed certain geometrical forms called Centro- baric} such that a body of that shape would attract another exactly as if the mass were concentrated at the centre of gravity, whether the forces act in a parallel manner or not ^ Thomson and Tait» Tr«U%H on NiUmnU Fhilotophy, voL i p. 594. i?i.] THE METHOD OF MEANS. 365 II Newton showed that uniform spheres of matter have this property, and this truth proved of the greatest importance in simplifying his calculations. But it is after all a purely hypothetical truth, because we can nowhere meet with, nor can we construct, a perfectly spherical and homogeneous body. The slightest iiTegularity or protrusion from the surface will destroy the rigorous correctness of the assump- tion. The spheroid, on the other hand, has no invariable centre at which its mass may always be regarded as con- centrated. The point from which its resultant attraction acts will move about according to the distance and posi- tion of the other attracting body, and it will only coincide with the centre as regards an infinitely distant body whose attractive forces may be considered as acting in parallel lines. Physicists speak familiarly of the poles of a magnet, and the term may be used with convenience. But, if we attach any definite meaning to the word, the poles are not the ends of the magnet, nor any fixed points within, but the variable points from which the resultants of all the forces exerted by the particles in the bar upon exterior magnetic particles may be considered as acting. The poles are, in short. Centres of Magnetic Forces ; but as those forces are never really parallel, these centres will vary in position according to the relative place of the object attracted. Only when we regard the magnet as attracting a very distant, or, strictly speaking, infinitely distant particle, do its centres become fixed points, situated in short magnets approximately at one-sixth of the whole length from each end of the bar. We have in the above instances of centres or poles of force sufficient examples of the mode in which the Fictitious Mean or Average is employed in physical science. The Precise Mean Remit, We now turn to that mode of employing the mean result which is analogous to the method of reversal, but which is brought into practice in a most extensive manner throughout many branches of physical science. We find the simplest possible case in the determination of the lati- tude of a place by observations of the Pole-star. Tycho ul I: i u I ; f ■ )1 if r^ (' 366 THE PRINCIPLES OP 80IKNCE. [OHAP. Brahe suggested that if the elevation of any circumpolar star were observed at its higher and lower passages across the meridian, half the sum of the elevations would be tlie latitude of the place, which is equal to the height of the pole. Such a star is as much above the pole at its highest passage, as it is below at its lowest, so that the mean must necessarily give the height of the pole itself free from doubt, except as regards incidental errors. The Pole-star is usually selected for the purpose of such observations because it describes the smallest circle, and is thus on the whole least affected by atmospheric refraction. Whenever several causes are in action, each of which at one time increases and at another time decreases the joint effect by equal quantities, we may apply this method and disentangle the effects. Thus the solar and lunar tides roll on in almost complete independence of each other. When the moon is new or full the solar tide coincides, or nearly so, with that caused by the moon, and the joint effect is the sum of the separate effects. When the moon is in quadrature, or half full, the two tides are acting in opposition, one raising and the other depressing the water, so that we observe oi3y the difference of the effects. We have in fact — Spring tide = lunar tide + solar tide ; Neap tide = lunar tide — solar tide. We have only then to add together the heights of tlie maximum spring tide and the minimum neap tide, and half the sum is the true height of the lunar tide. Half the difference of the spring and neap tides on the other hand gives the solar tide. Effects of very small amount may be detected with great approach to certainty among much greater fluctua- tions, provided that we have a series of observations suf- ciently immerous and long continued to enable us to balance all the larger effects against each other. For this purpose the observations should be continued over at least one complete cycle, in which the effects run through all their variations, and return exactly to the same relative positions as at the commencement. If casual or irregular disturbing causes exist, we should probably require many such cycles of results to render their effect inappreciable. We obtain the desired result by taking the mean of all the xvij THE METHOD OF MEANS. 367 observations in which a cause acts positively, and the mean of all in which it acts negatively. Half the diffe- I'ence of these means will give the effect of the cause in question, provided that no other effect happens to vary in tiie same period or nearly so. Since the moon causes a movement of the ocean, it is evident that its attraction must have some effect upon the atmosphere. The laws of atmospheric tides were investi- gated by Laplace, but as it would be impracticable by theory to calculate their amounts we can only determine them by observation, as Laplace predicted that they would one day be determined.^ But the oscillations of the barometer thus caused are far smaller than the oscillations due to several other causes. Storms, hurricanes, or changes of weather produce movements of the barometer some- times as much as a thousand times as great as the tides in question. There are also regular daily, yearly, or other fluctuations, all greater than the desired quantity. To detect and measure the atmospheric tide it was desirable that observations should be made in a place as free as possible from irregular disturbances. On this account several long series of observations were made at St. Helena, where the barometer is far more regular in its movements than in a continental clinrnte. The effect of the moon's attraction was then detected by taking the mean of all the readings when the moon was on the me- ridian and the similar mean when she was on the horizon. The difference of these means was found to be only 00^6$, yet it was possible to discover even the variation of this tide according as the moon was nearer to or further from the earth, though this difference was only 00056 inch.* It is -quite evident that such minute effects could never be discovered in a purely empirical manner. Having no information but the series of observations before us, we could have no clue as to the mode of grouping them which would give so small a difference. In applying this method of means in an extensive manner we must gene- rally then have d priori knowledge as to the periods at which a cause will act in one direction or the other. * E»»a% PhUosophiqiie sur lei ProbabiliUs, pp. 49, 50. • Grant, Hitiory of Physical Astronomy, p. 163. iq I 1 j ! -i ill / 368 THE PRINCIPLES OF SCIENCE. [OHAP. We are sometimes able to eliminate fluctuations and take a mean result by purely mechanical an-angements. The daily variations of temperature, for instance, become imperceptible one or two feet below the surface of the earth, so that a thermometer placed with its bulb at that depth gives very nearly the true daily mean temperature. At a depth of twenty feet even the yearly fluctuations are nearly eflaced, and the thermometer stands a little above the true mean temperature of the locality. In registering the rise and fall of the tide by a tide-gauge, it is desirable to avoid the oscillations arising from surface waves, which is very readily accomplished by placing the float in a cis- tern communicating by a small hole with the sea. Only a general rise or fall of the level is then perceptible, just as in the marine barometer the narrow tube prevents any casual fluctuations and allows only a continued change of pressure to manifest itself. Determination of the Zero point. In many important observations the chief difficulty con- sists in defining exactly the zero point from which we are to measure. We can point a telescope with great pre- cision to a star and can measure to a second of arc the angle through which the telescope is raised or lowered ; but all this precision will be useless unless we know exactly the centre point of the heavens from which we measure, or, what comes to the same thing, the horizontal line 90° distant from it. Since the true horizon has reference to the figure of the earth at the place of observation, we can only determine it by the direction of gravity, as marked either by the plumb-line or the surface of a liquid. The question resolves itself then into the most accurate mode of observing the direction of gravity, and as the plumb-line has long been found hopelessly inaccurate, astronomers generally employ the surface of mercury in repose as the criterion of horizon- tality. They ingeniously observe the direction of the surface by making a star the index. From the laws of reflection it follows that the angle between the direct ray from a star and that reflected from a surface of mercury will be exactly double the angle between the xtl] THE METHOD OP MEANS. 369 surface and the direct ray from the star. Hence the horizontal or zero point is the mean between the apparent place of any star or other very distant object and its reflection in mercury. A plumb-line is perpendicular, or a liquid surface is horizontal only in an approximate sense ; for any irregu- larity of the surface of the earth, a mountain, or even a house must cause some deviation by its attracting power. To detect such deviation might seem very difficult, because every other plumb-line or liquid surface would be equally affected by gravity. Nevertheless it can be detected ; for if we place one plumb-line to the north of a mountain, and another to the south, they will be about equally deflected in opposite directions, and if by observations of the same star we can measure the angle between the plumb-lines, half the inclination will be the deviation of either, after allowance has been made for the inclination due to the difference of latitude of the two places of observation. By this mode of observation applied to the mountain Schiehal- lion the deviation of the plumb-line was accurately measured by Maskelyne, and thus a comparison instituted between the attractive forces of the mountain and the whole globe, which led to a probable estimate of the earth's density. In some cases it is actuaUy better to determine the zero point by the average of equally diverging quantities than by direct observation. In delicate weighmgs by a chemical balance it is requisite to ascertain exactly the point at which the beam comes to rest, and when standard weights are being compared the position of the beam is ascertained by a carefully divided sciale viewed through a microscope. But when the beam is just coming to rest, friction, small impediments or other accidental causes may readily ob- struct it, because it is near the point at which the force of stability becomes infinitely small Hence it is found better to let the beam vibrate and observe the terminal points of the vibrations. The mean between two extreme points will nearly indicate the position of rest Friction and the resistance of air tend to reduce the vibrations, so that this mean will be eiToneous by half the amount oi this effect during a half vibratioa But by taking several ob- servations we may determine this retardation and allow for it Thus if a, ft, c be the readings of the terminal B B ill ■ ■ 1 1 I It ;• ft .170 THE 1>RINCI?LKS OP SCIENOK. tcitAf. points of three excursions of the beam from the zero of the scale, then J (a + ft) will be about as much erroneous in one direction as ^ (ft + c) in the other, so that the mean of these two means, or J (a + 2 ft + c), will be exceedingly near to the point of rest^ A still closer approximation may be made by taking four readings and reducing them by the formula J(a + 2ft4-2 c '\- d). The accuracy of Baily's experiments, directed to deter- mine the density of the earth, entirely depended upon this mode of observing oscillations. The balls whose gmvi- tation was measured were so delicately suspended by a torsion balance that they never came to rest The extreme points of the oscillations were observed both when the heavy leaden attracting ball was on one side and on the other. The difference of the mean points when the leaden ball was on the right hand and that when it was on the left hand gave double the amount of the deflection. A beautiful instance of avoiding the use of a zero point is found in Mr. K J. Stone's observations on the radiant heat of the fixed stars. The difficulty of these obsei-vations arose from the comparatively great amounts of heat which were sent into the telescope from the atmosphere, and which were sufficient to disguise almost entirely the feeble heat rays of a star. But Mr. Stone fixed at the focus of his telescope a double thermo-electric pile of which the two parts were reversed in order. Now any disturbance of temperature which acted uniformly upon both piles pro- duced no effect upon the galvanometer needle, and when the rays of the star were made to fall alternately upon one pile and the other, the total amount of the deflection represented double the heating power of the star. Thus Mr. Stone was able to detect with much certainty a heating effect of the star Arcturus, which even when concentrated by the telescope amounted only to o°02 Fahr., and which represents a heating effect of the direct ray of only about o°ocxxx)i37 Fahr., equivalent to the heat which would be received from a three-inch cubic vessel full of boiling water at the distance of 400 yards.* It is probable that * Gaues, Taylor's ScierUi/ic Memoirt, vol. ii. p. 43, &c. * Proeudings of the Moyal Society t vol. xviii. p. 159 (Jan. 13, 1870). Pkilosophical Magaziru (4th Series), voL xxxix. p. 376. XVI.] THE METHOD OF MEA.NS. 871 Mr. Stone's arrangement of the pile might be usefully employed in other delicate thermometric experiments subject to considerable disturbing influences. Determination of Maximum Points. We employ the method of means in a certain number of observations directed to determine the moment at which a phenomenon reaches its highest point in quantity. In noting the place of a fixed star at a given time there is no difficulty in ascertaining the point to be observed, for a star in a good telescope presents an exceedingly small disc. In observing a nebulous body which from a bright centre fades gradually away on all sides, it will not be possible to select with certainty the middle point. In many such cases the best method is not to select arbitrarily the sup- posed middle point, but points of equal brightness on either side, and then take the mean of the observations of these two points for the centre. As a general rule, a variable quantity in reaching its maximum increases at a less and less rate, and after passing the highest point begins to decrease by insensible degrees. The maximum may indeed be defined as that point at which the increase or decrease is null. Hence it will usually be the most indefinite point, and if we can accurately measure the phenomenon we shall best determine the place of the maximum by determining points on either side at which the ordinates are equal. There is moreover this advantage in the method that several points may be determined with the corresponding ones on the other side, and the mean of the whole taken as the true place of the maximum. But this method entirely depends upon the existence of sym- metry in the curve, so that of two equal ordinates one shall be as far on one side of the maximum as the other is on the other side. The method fails when other laws of variation prevail In tidal observations great difficulty is encountered in fixing the moment of high water, because the rate at which the water is then rising or falling, is almost impercep- tible. Whewell proposed, therefore, to note the time at which the water passes a fixed point somewhat below the maximum both in rising; and falling, and take the mean BB 2 ilill in I » •I ( V il^( r^ 872 THE PRINCIPLES OP SCIENCl!.. [cnAF. time as that of high water. But this mode of proceeding unfortunately does not give a correct result, because the tide follows different laws in rising and in falling. There is a difficulty again in selecting the highest spring tide, another object of much importance in tidology. Laplace discovered that the tide of the second day preceding the conjunction of the sun and moon is nearly equal to that of the fifth day following; and, believing that the increase and decrease of the tides proceeded in a nearly symmetrical manner, he decided that the highest tide would occur about thirty-six hours after the conjunction, that is half-way between the second day before and the fifth day after.* This method is also emplo}'ed in determining the time of passage of the middle or densest point of a stream of meteors. The earth takes two or three days in passing completely through the November stream ; but astronomers need for their calculations to have some definite point fixed within a few minutes if possible. When near to the middle they observe the numbers of meteors which come within the sphere of vision in each half hour, or quartei hour, and then, assuming that the law of variation is symmetrical, they select a moment for the passage of the centre equidistant between times of equal frequency. The eclipses of Jupiter's satellites are not only of great interest as regards the motions of the satellites themselves, but were, and perhaps still are, of use in determining longitudes, because they are events occurring at fixed moments of absolute time, and visible in all parts of the planetary system at the same time, allowance being made for the interval occupied by the light in travelling. But, as is explained by Herschel,* the moment of the event is wanting in definiteness, partly because the long cone of Jupiter's shadow is surrounded by a penumbra, and partly because the satellite has itself a sensible disc, and takes time in entering the shadow. Different obseiTers using different telescopes would usually select different moments for that of the eclipse. But the increase of light in the emersion will proceed according to a law the reverse of that observed in the immersion, so that if an observer notes ' Airy On Tides and Waves, Encycl. Metrop. pp. 364* — 366*. * OuiUnu qf Astronomy^ 4th edition, { 538 XVI.3 THE METHOD OF MEANS. 373 the time of both events with the same telescope, he will be as much too soon in one observation as he is too late in the other, and the mean moment of the two observations will represent with considerable accuracy the time when the satellite is in the middle of the shadow. Error of judg- ment of the observer is thus eliminated, provided that he takes care to act at the emei-sion as he did at the immersion. I I CHAPTER XV 11. THE LAW OF EKROR. \ h / ( To bring error itself under law might seem beyond human power. He who errs surely diverges from law, and it might be deemed hopeless out of error to draw truth. One of the most remarkable achievements of the human intel- lect is the establishment of a general theory which not only enables us among discrepant results to approximate to the truth, but to assign the degree of probability which fairly attaches to this conclusion. It would be a mistake indeed to suppose tliat this law is necessarily the best guide under all circumstances. Every measuring instru- ment and every form of experiment may have its own special law of error ; there may in one instrument be a tendency in one direction and in another in the opposite direction. Every process has its peculiar liabilities to disturbance, and we are never i-elieved from the necessity of providing against special difficulties. The general Law of Enx)r is the best guide only when we have exhausted all other means of approximation, and still find discrepancies, which are due to unknown causes. We must treat such residual differences in some way or other, since they wiU ^ occur in all accurate experiments, and as their origin is assumed to be unknown, there is no reason why we should treat them differently in different cases. Accordingly the ultimate Law of Error must be a uniform and general one. It is perfectly recognised by mathematicians that in each case a special Law of Error may exist, and should be discovered if possible. "Nothing can be more unlikely t^an that the errors committed in all classes of observa- Cll. XVII.] THE LAW OF ERROR. 376 tions should follow the same law," ^ and the special Laws of Error which will apply to certain instmments, as for in- stance the repeating circle, have been investigated by Bi*avais.2 He concludes that every distinct cause of error gives rise to a curve of possibility of errors, which may have any form, — a curve which we may either be able or unable to discover, and which in the first case may be determined by d priori considerations on the peculiar nature of this cause, or which may be determined d posteriori by observation. Whenever it is practicable and worth the labour, we ought to investigate these special conditions of error ; nevertheless, when there are a great number of different sources of minute error, the general resultant will always tend to obey that general law which we are about to consider. Establishment of the Law of Erroi: Mathematicians agree far better as to the form of the Law of Error than they do as to the manner in which it can be deduced and proved. They agree that among a number of discrepant results of observation, that mean quantity is probably the best approximation to the truth which makes the sum of the squares of the errors as small as possible. But there are three prin cipaLway s i n which this ^ law J iasJ^gen^arriveH" at Tespgctive^,l)y Grauss^ by LajJace^'Sid^.uetelet.^SSIE^ ^ir'Jo I m Herscbel. Gauss proceeds^uch upon assumptions^ Herschel rests*" upon geometrical considerations ; while Laplace and Quetelet regard the Law of Error as a development of the doctrine of combinations. A number of other mathematicians, such as Adrain of New Brunswick, Bessol, Ivory, Donkin, Leslie Ellis, Tait, and Crofton have either attempted independent proofs or have modified or commented on those here to be described. For full accounts of the literature of the subject the reader should refer either to Mr. Todhunter's History of the Theory of Prohahility or to the able memoir of Mr. J. W. L Glaisher.8 ' Philosophical Magazine^ 3rd Series, vol xxxvii. p. 324. * Letters on the Theory of FrohabilitieSy by Quetelet, translated by 0. G. Downes, Notes to Letter XXVL pp. 286—295. ^ On the Law of Facility of Errors of OhservationSj and on th* Method of Least Squares^ Memoirs of the Royal Astroaomical Society, Yol. xxxix. p. 75. ^'1 1 o I S76 THE PRINCIPLES OP SCIENCE. [OBAP. ^■^ H According to Gauss the Law of Error expresses the comparative probability of errors of various magnitude, and partly from experience, partly from d ^^rixyri considera- tions, we may readily lay down certain conditions to which the law will certainly conform. It may fairly be assumed as a first principle to guide us in the selection of the law, that large errors will be far less frequent and probable than small ones. We know that very large errors are almost impossible, so that the probability must rapidly decrease as the amount of the error increases. A second principle is that positive and n^ative errors shall be equally probable, which may certainly be assumed, because we are supposed to be devoid of any knowledge as to the causes of the residual errors. It follows that the proba- bility of the error must be a function of an even power of the magnitude, that is of the square, or the fourth power, or the sixth power, otherwise the probability of the same amount of error would vary according as the error was positive or negative. The even powers ai", a?*, ««, &c., are always intrinsically positive, whether x be positive or negativa There is no <i ^grwri reason why one rather than another of these even powers should be selected. Gauss himself allows that the fourth or sixth power would fulfil the conditions as well as the second ; * but in the absence of any theoretical reasons we shoultl prefer the second power, because it leads to formulae of great comparative simplicity. Did the Law of Error necessitate the use of the higher powers of the error, the complexity of the necessary calculations would much reduce the utility of the theory. By mathematical reasoning which it would be unde- sirable to attempt to follow in this book, it is shown that under these conditions, the facility of occurrence, or in other words, the probability of error is expressed by a function of the general form c"** •", in which x repre- sents the variable amount of errors. From this law, to be more fully described in the following sections, it at once follows that the most probable result of any observa- » Mithfidt dea Moindres Carrit, Mivwire* tur la CombimnUon dei OWohoiM, par Ch. Fr Oauss. Ttaduit m Fran^aU par J. 9«r<rani, Pans, 1855, pp. 6, 133, &c ^ V ^' ivii.j THE LAW OF ERROR. 377 tions is that which makes the sum of the squares of the consequent errors the least possible. Let a, h, c, &C., be the results of observation, and x the quantity selected as the most probable, that is the most free from unknown errors : then we must determine x so that (a - ac)* + (J - »)* + (c - a;)2 + . . . . shall be the least possible quantity. Thus we arrive at the celebrated MdhoiJL^^f-^LMst Sqvures, as it is usually called, which appears to have been first distinctly put in practice by Gauss in 1795, while Legendre first published in 1806 an account of the process in his work, entitled, Nouvelles Mithodes pour la Determination des Orhites des CorrUtes. It is worthy of notice, however, that Roger Cotes had long previously recommended a method of equivalent nature in his tract. " Estimatio Erroris in Mixta Mathesl" ^ Her8chel*8 Geometrical Proof, A second way of arriving at the Law of Error was proposed by Herschel, and although only applicable to geometrical cases, it is remarkable as showing that from whatever point of view we regard the subject, the same principle will be detected. Aft/Cr assuming that some general law must exist, and that it is subject to the principles of probability, hfe supposes that a ball is dropped from a high point with the intention that it shall strike a given mark on a horizontal plane. In the absence of any known causes of deviation it will either strike that mark, or, as is infinitely more probable, diverge from it by an amount which we must regard as error of unknown origin. Now, to quote the words of Herschel,^ " the probability of that error is the unknown function of its square, i.e. of the sum of the squares of its deviations in any two rectangular directions. Now, the probability of any deviation depending solely on its magnitude, and not on its direction, it follows that the probability of each of these rectangular deviations must be the same function of its square. And since the observed oblique deviation is * De Morgan, Penny Cyclopaedia, art. Least Squares, * Edinburgh Bevitu), July 1850, vol. xciL p. 17. Reprinted EsgaySy p. 399. This method of demonstration is discussed by Boole, Tram- Qdiom of Royal Society qf Edinburgh^ voL xxi. pp. 627 — 630, ill ! 1 .1(1 378 THE PRINCIPLES OF SCIENCE. [chap. .1 equivalent to the two rectaDgular ones, supposed concur- rent, and which are essentially independent of one another, and is, therefore, a compound event of which they are the simple independent constituents, therefore its probability will be the product of their separate probabilities. Thus the form of our unknown function comes to be determined from this condition, viz., that the product of such functions of two independent elements is equal to the same function of their sum. But it is shown in every work on algebra that this property is the peculiar characteristic of, and belongs only to, the exponential or antilogarithmic function. . This, then, is the function of the square of the error, which vf expresses the probability of committing that error. That probability decreases, therefore, in geometrical progression, as the square of the error increases in arithmetical." Laplace s and Queteht*8 Proof of the Law, However much presumption the modes of determining the Law of Error, already described, may give in favour of the law usually adopted, it is difficult to feel that the arguments are satisfactory. The law adopted is chosen rather on the grounds of convenience and pkusibDity, than because it can be seen to be the necessary law. We can however approach the subject from an entirely different point of view, and yet get to the same result. Let us assume that a particular observation is subject to four chances of error, each of which will increase the result one inch if it occurs. Each of these errors is to be regarded as an event independent of the rest and we can therefore assign, by the theory of probability, the compara- tive probability and frequency of ea^h conjunction of errors. From the Arithmetical Triangle (pp. 182-188) we learn that no error at all can happen only in one way ; an error of one inch can happen in 4 ways ; and the ways of happening of errors of 2, 3 and 4 inches respectively, will be 6, 4 and I in number. We may infer that the error of two inches is the most likely to occur, and will occur in the long run in six cases out of sixteen. Errors of one and three inches will be equally likely, but will occur less frequently ; while no eiTor at all or one of four inches will be a comparatively XVII.) THE LAW OF ERROR. 379 > — , — rare occurrence. If we now suppose the errors to act as often in one direction as the other, the effect will be to alter the average error by the amount of two inches, and we shall have the following results : — Negative error of 2 inches i way Negative error of i inch 4 ways. No error at all 6 ways. Positive error of i inch ...... 4 ways. Positive error of 2 inches i way. We may now imagine the number of causes of error increased and the amount of each error decreased, and the aiithmetical triangle will give us the frequency of the re- sulting errors. Thus if there be five positive causes of error and five negative causes, the following table shows the numbers of errors of various amount which will be the result : — Direction of Error. Positive Error. Negative Error. Amount uf EUror. 5. 4. 3. a. « 25a i> 3> 3> 4. 5 Number of such ESrrors. I, 10, 45, 120, 210 aio, lao, 45, lo, i It is plain that from such numbers I can ascertain the probability of any particular amount of en'or under the conditions supposed. The probability of a positive 210 error of exactly one inph is > in which fraction the •^ 1024 numerator is the number of combinations giving one inch positive error, and the denominator the whole number of possible errors of all magnitudes. I can also, by adding together the appropriate numbers get the pro- bability of an error not exceeding a certain amount. Thus the probability of an error of three inches or less, positive or negative, is a fraction whose numerator is the sum of 45 4- 120 -f- 210 4- 252 + 210 + 120 -{■ 45, and the deno- minator, as before, giving the result 1002 1024* We may see at once that, according to these principles, the probability of small errors is far greater than of large ones : the odds are 1002 to 22t or more than 45 to i that the error will not / / I II ^vh t 980 THB PRINCIPLES OP SCIBNOE. [OBAP. exceed three inches ; and the odds are 1022 to 2 against the occurrence of the greatest possible error of five inches. If any case should arise in which the observer knows the number and magnitude of the chief errors which may occur, he ought certainly to calculate from the Arith- metical Triangle the special Law of Error which would apply. But the general law, of which we are in search, is to be used in the dark, when we have no knowledge whatever of the sources of error. To assume any special number of causes of error is then an arbitrary pi-oceeding, . rand mathematicians have chosen the least arbitrary course ' 'ol imagining the existence of an infinite number of in- |fibDitely small errors, just as, in the inverse method of /probabilities, an infinite number of infinitely improbable 'hypotheses were submitted to calculation (p. 255). The reasons in favour of this choice are of several different kinds. 1. It cannot be denied that there may exist infinitely numerous causes of error in any act of observation. 2. The law resulting from the hypothesis of a moderate number of causes of error, does not appreciably differ from that given by the hypothesis of an infiiute number of causes of error. 3. We gain by the hypothesis of infinity a general law capable of ready calculation, and applicable by uniform rules to all problems. 4. This law, when tested by comparison with extensive series of observations, is strikingly verified, as will be shown in a later section. When we imagine the existence of any large number of causes of error, for instance one hundred, the numbers of combinations become impracticably large, as may be seen to be the case from a glance at the Arithmetical Triangle, which proceeds only up to the seventeenth line. Quetelet, by suitable abbreviating processes, calculated out a table of probability of errors on the hypothesis of one thousand distinct causes;* but mathematicians have generally proceeded on the hypothesis of infinity, and then, by the devices of analysis, have substituted a general law of easy » Ldters on the Theory of ProbdbiliHet, Letter XV. and Appendix, aote pp. 256 -266. I xni.] THE LAW OP ERROR m treatment. In mathematical works upon the subject, it is shown that the standard Law of Error is expressed in the formula in which x is the amount of the error, Y the maximum ordinat'C of the curve of error, and c a number constant for each series of observations, and expressing the amount of the tendency to error, varying between one series of observations and another. The letter e is the mathematical constant, the sum of ratios between the numbers of permu- tations and combinations, previously referred to (p. 330). To show the close correspondence of this general law with the special law which might be derived from the supposition of a moderate number of causes of error, I have in the accompanying figure drawn a -5 -I -B -I -1 -t curved line representing accurately the variation of y when X in the above formula is taken equal o, -, i, - 2, &c., positive or negative, the arbitmry quantit^s Y and c being each assumed equal to unity, in order to simplify the calculations. In the same figure are inserted eleven dots, whose heights above the base line are proportional to tJie numbers in the eleventh line of the Arithmetical Triangle, thus representing the comparative probabilities of errors of various amounts arising tix>m ten equal causes fill I I \l H I Sd2 THE PmNCTl>LES OF SCIENCK [chap. of error. The correspondence of the general and the special Law of Error is almost as close as can be exhibited in the figure, and the assumption of a greater number of equal causes of error would render the correspondence far more close. It may be explained that the ordinates NM, nw, n'm\ represent values of y in the equation expressing the I^aw of Error. The occurrence of any one definite amount of error is infinitely improbable, because an infinite number of such ordinates might l)e drawn. But the probability of an error occuiTing between certain limits is finite, and is represented by a portion of the area of the curve. Thus the probability that an error, positive or negative, not exceed- ing unity will occur, is represented by the area Mmnn'm', in short, by the area standing upon the line nn. Since every observation must either have some definite error or none at all, it follows that the whole area of the curve should be considered as the unit expressing certainty, and the probability of an error falling between particular limits will then be expressed by the ratio which the area of the curve between those limits bears to the whole area of the curve. The mere fact that the Law of Error allows of the posst ble existence of qtyots of every assignable amount showa that it is only approximately true. We may fairly say that in measuring a mile it would be impossible to commit an error of a hundred miles, and the length of life would never allow of our committing an error of one million miles. Nevertheless the general Law of Error would assign a probability for an error of that amount or more, but so small a probability as to be utterly inconsiderable and almost inconceivable. All that can, or in fact need, be said in defence of the law is, that it may be made to re- present the errors in any special case to a very close approximation, and that the probability of large and prac- tically impossible errors, as given by the law, will be so small as to be entirely inconsiderable. And as we are dealing with error itself, and our results pretend to nothing more than approximation and probability, an indefinitely small error in our process of approximation is of no import- ance whatever. xtii I THE LAW OF ERROR m Logical Origin of the Lata of Error, It is worthy of notice that this Law of Error, abstruse though the subject may seem, is really founded upon the simplest principles. It arises entirely out of the difference between permutations and combinations, a subject upon which I may seem to have dwelt with unnecessary prolixity in previous pages (pp. 170, 189). The order in which we add quantities together does not affect the amount of the sum, so that if there be three positive and five negative causes of error in operation, it does not matter in which order they are considered as acting. They may be inter- mixed in any arrangement, and yet the result will be the same. The reader should not fail to notice how laws or principles which appeared to be absurdly simple and evident when first noticed, reappear in the most complicated and mysterious processes of scientific method. The funda- mental Laws of Identity and Difference gave rise to the Logical Alphabet which, after abstracting the character of the differences, led to the Arithmetical Triangle. Th6 Law of Error is defined by an infinitely high line of that triangle, and the law proves that the mean is the most pro- bable result, and that divergencies from the mean become much less probable as they increase in amount. Now the comparative greatness of the numbers towards the middle of each line of the- Arithmetical Triangle is entirely due to the indifference of order in space or time, which was first prominently pointed out as a condition of logical re- lations, and the symbols indicating them (pp. 32-35), and which was afterwards shown to attach equally to numerical symbols, the derivatives of logical terms (p. 160). Verification of the Law of Error. The theory of error which we have been considering rests entirely upon an assumption, namely that when known sources of disturbances are allowed for, there yet remain an indefinite, possibly an infinite number of other minute sources of error, which will as often produce ex- cess as deficiency. Granting this assumption, the Law of Error must be as it is usually taken to be, and there is DO more need to verify it empirically than to test the truth I i( .( * f 1^ l\ 384 THE PRINCIPLES OF SCIENCK [CBAT. of one of Euclid's propositions mechanically. Neverthe- less, it is an interesting occupation to verify even the pro- positions of geometry, and it is still more instructive to try whether a large number of observations will justify our assumption of the Law of Error. Encke has given an excellent instance of the correspond- ence of theory with experience, in the case of observations of the differences of Right Ascension of the sun and two stars, namely a Aquilse and a Canis minoris. The obser- vations were 470 in number, and were made by Bradley and reduced by Bessel, who found the probable error of the final result to be only about one-fourth part of a second (0*2637). He then compared the numbers of errors of each magnitude from o* i second upwards, as actually given by the observations, with what should occur according to the Iaw of Error. The results were as follow : — ^ of a Moond. Namber of erron of each magnitado Moording to Obaervtfcioii. Theory. 00 to o't I ., •« • .. -3 '3 - '4 ♦ - *i •5 ., « 7 f» •8 ,. 9 -9 M « .» » bore < 8 I 1 64 •4 ■S 9 S s The reader will remark that the correspondencif is very close, except as regards larger errors, which are excessive in practica It is one objection, indeed, to the theory of error, that, being expressed in a continuous mathematical function, it contemplates the existence of errors of every magnitude, such as could not practically occur ; yet in this case the theory seems to under-estimate the number of large errors. 1 Encke, On the Method 0/ Lead iSgitarei, Taylor's ScietU^fic Mumoinf vol ii. pp. 338, 339. XVII.J THE LAW OF ERROR. 385 Another comparison of the law with observation was made by Quetelet, who investigated the errors of 487 determi- nations in time of the Right Ascension of the Pole-Star made at Greenwich during the four years 1836-39. These observations, although carefully corrected for all known causes of error, as well as for nutation, precession, &c., are yet of course found to differ, and being classified as regards intervals of one-half second of time, and then pro- portionately increased in number, so that their sum may be one thousand, give the following results as compared with what Quetelet's theory would lead us to expect : — * Magnitude of error in truths of a second. Number of Errors Magnitude of error in tcntlis of 4 second. Number of errors »»3r Observation. Theory. by Observation. Theory. o'o + 05 + ro + 1-5 + a + •5 + 3-0 itfS •48 199 78 33 10 9 '63 »47 113 7a 40 to —05 — I'O -I 5 — 20 -2*5 —30 — 35 126 74 43 as 12 3 153 131 82 46 3S 10 4 In this instance also the correspondence is satisfactory, but the divergence between theory and fact is in the opposite direction to that discovered in the former comparison, the larger errors being less frequent than theory would indi- cate. It will be noticed that Quetelet's theoretical results are not symmetrical The Probable Mean Result. One immediate result of the Law of Error, as thus stated, is that the mean result is the most probable one ; and when there is only a single variable this mean is found by the familiar arithmetical process. An unfor- tunate error has crept into several works which allude to this subject. Mill, in treating of the " Elimination of Chance,*' remarks in a note * that " the mean is spoken of ' Quetelet, Letters on the Tluory of Probabilities, translated by Downes, Letter XIX. p. 88. See also Galtoii's Ilereditary Oenxus, p. ^79. System of Logic, bk. iiL chap. 17, § 3. 5tb ed. vol. ii. p. 56. G C \\\ i'J 1 f II ( t it M t 386 THE PRINCIPLES OF SCIENCE. [chap. as if it were exactly the same thing as the average. But the mean, for purposes of inductive inquiry, is not the average, or arithmetical mean, though in a familiar illus- tration of the theory the difference may be disregarded.** He goes on to say that, according to mathematical princi- ples, the most probable result is that for which the sums of the squares of the deviations is the least possible. It seems probable that Mill and other writers were misled by Whewell, who says^ that "Tlie method of least squares is in fact a method of means, but with some peculiar characters. . . . The method proceeds upon this supposition : that all errors are not equally probable, but that small errors are more probable than large ones." He adds that this method '* removes much that is arbitrary in the method of means." It is strange to find a mathe- matician like Whewell making such remarks, when there is no doubt whatever that the Method of Means is only an application of the Method of Least Squares. They are, in fact, the same method, except that the latter method may be applied to cases where two or more quantities have to be determined at the same time. Lubbock and Drink- water say,* " If only one quantity has to be determined, this method evidently resolves itself into taking the mean of all the values given by observation." Encke says,' that the expression for the probability of an error " not only contains in itself the principle of the arithmetical mean, but depends so immediately upon it, that for all those magnitudes for which the arithmetical mean holds good in the simple cases in which it is principally applied, no other law of probability can be assumed than that which is expressed by this formula." The Probahle Error of Results. When we draw a conclusion from the numerical results of observations we ought not to consider it suf- ficient, in cases of importance, to content ourselves with finding the simple mean and treating it as true. We ought also to ascertain what is the degree of confidence * Pkiloiophy of the Inductive Sciences, 2nd ed. vol. ii. pp. 408, 409. * Euay <m FrohahUity, Useful Knowledge Society, 1833, p. 41. * Taylor's Scientific Memoiri^ vol. ii. p. 333. aviij THE LAW OF ERROR. 381 we may place in this mean, and our confidence should be measured by the degree of concurrence of the observations from which it is derived. In some cases the mean may be approximately certain and accurate. In other cases it may really be worth little or nothing. The Law of Error enables us to give exact expression to the degree of con- fidence proi)er in any case ; for it shows how to calculate the probability of a divergence of any amount from the mean, and we can thence ascertain the probability that the mean in question is within a certain distance from the true number. The probable error is taken by mathema- ticians to mean the limits within which it is as likely as not that the truth will fall. Thus if 5 45 be the mean of all the determinations of the density of the earth, and '20 be approximately the probable error, the meaning is that the probability of the real density of the earth falling be- tween 5 2 5 and 5 65 is J. Any other limits might have been selected at will. We might calculate the limits within which it was one hundred or one thousand to one that the truth would fall ; but there is a convention to take the even odds one to one, as the quantity of proba- bility of which the limits are to be estimated. Many books on probability give rules for making the calculations, but as, in the progress of science, persons ought to become more familiar with these processes, I propose to repeat the rules here and illustrate their use. The calculations, when made in accordance with the directions, involve none but arithmetic or logar- ithmic operations. The following are the rules for treating a mean result, so as thoroughly to ascertain its trustworthiness. 1. Draw the mean of all the observed results. 2. Find the excess or defect, that is, the error of each result from the mean. 3. Square each of these reputed errors. 4. Add together all these squares of the errors, which are of course all positive. 5. Divide by one less than the number of observations. This gives the sqiuire of the mean error, 6. Take the square root of the last result ; it is the mean error of a single observation. 7. Divide now by the square root of tlie number of cc 2 i \i w *y i' i THE PRINCIPLES OP SCIENCE. [chap. observations, and we get the mean error of tJie mean result. 8. Lastly, multiply by the natural constant 06745 (or approximately by 0*674, or even by J), and we arrive at the probable error of the mean result Suppose, for instance, that five measurements of the height of a hill, by the barometer or otherwise, have given the numbers of feet as 293, 301, 306, 307, 313 ; we want to know the probable error of the mean, namely 304. Now the differences between this mean and the above numbers, paying no regard to direction, are ii, 3, 2, 3, 9; their squares are 121, 9, 4, 9, 81, and the sum of the squares of the errors consequently 224. The number of observa- tions being 5, we divide by i less, or 4, getting 56. This is the square of the mean error, and taking its square root we have 7*48 (say 7 J), the mean error of a single obser- vation. Dividing by 2236, the square root of 5, the number of observations, we find the mean error of the mean result to be 3*35, or say 3 J, and lastly, multiplying by •6745, we arrive at the probable error of the mean result, which is found to be 2259, or say 2J. The meaning of this is that the probability is one half, or the odds are even that the true height of the mountain lies between 301} and 306J feet. We have thus an exact measure of the degree of credibility of our mean result, which mean indicates the most likely point for the truth to fall upon. The reader should observe that as the object in these calculations is only to gain a notion of the degree of con- fidence with which we view the mean, there is no real use in caiTying the calculations to any great degree of pre- cision ; and whenever the neglect of decimal fractions, or 6ven the slight alteration of a number, will much abbre- viate the computations, it may be fearlessly done, except in cases of high importance and precision. Brodie has shown how the law of error may be usefully applied in chemical investigations, and some illustrations of its em- ployment may be found in his paper.* The experiments of Benzenberg to detect the revolution of the earth, by the deviation of a ball from the perpen- ' Philosophical Tranaactwns, 1873, P* ^3* XTII.] THE LAW OF ERROR. 389 dicular line in falling down a deep pit, have been cited by Encke^ as an interesting illustration of the Law of Error. The mean deviation was 5 086 lines, and its probable error was calculated by Encke to be not more than -950 line, that is, the odds were even that the true result lay between 4" 1 36 and 6036. As the deviation, according to astrono- mical theory, should be 46 lines, which lies well within the limits, we may consider that the experiments are consistent with the Copemican system of the universe. It will of course be understood that the probable error has regard only to those causes of errors which in the lonf« run act as much in one direction as another ; it takes no account of constant errors. The true result accordingly will often fall far beyond the limits of probable error, owing to some considerable constant error or errors, of the ex- istence of which we are unaware. Bisection of the Mean Besult. We ought always to bear in mind that the mean of any series of observations is the best, that is, the most probable approximation to the truth, only in the absence of know-s ledge to the contrary. The selection of the mean restst entirely upon the probability that unknown causes of eiTor will in the long run fall as often in one direction as the opposite, so that in drawing the mean they will balance each other. If we have any reason to suppose that there exists a tendency to error in one direction rather than the other, then to choose the mean would be to iguore that tendency. We may certainly approximate to the length of the circumference of a circle, by taking the mean of the perimeters of inscribed and circumscribed polygons of an equal and large number of sides. The length of the cir- cular line undoubtedly lies between the lengths of the two perimeters, but it does not follow that the mean is the best approximation. It may in fact be shown that the circumference of the circle is very nearly equal to the perimeter of the inscribed polygon, together with one -third part of the difference between the inscribed and circum- scribed polygons of the same number of sides. Having • Taylor's Scimtxfie Memoirt, vol. ii pp. 330, 347, &c. Il I I 1 '•il.i . Il t '. 890 THE PRINCIPLES OF SCIENCE. [OBAP. this knowledge, we ought of course to act upon it, instead of trusting to probability. We may often perceive that a series of measurements tends towards an extreme limit rather than towards a mean. In endeavouring to obtain a correct estimate of the apparent diameter of tlie brightest fixed stars, we find a continuous diminution in estimates as the powers of observation increased. Kepler assigned to Sirius an apparent diameter of 240 seconds ; Tycho Brahe made it 126; Gassendi 10 seconds; Galileo, Hevelius, and J. Cassini, 5 or 6 seconds. Halley, Michell, and subsequently Sir W. Herscliel came to the conclusion that the brightest stars in the heavens could not have real discs of a second, and were probably much less in diameter. It would of course be absurd to take the mean of quantities which differ more than 240 times; and as the tendency has always been to smaller estimates, there is a considerable presumption in favour of the smallest.^ In many experiments and measurements we know that there is a preponderating tendency to error in one direc- tion. The readings of a thermometer tend to rise as the age of the instrument increases, and no drawing of means will correct this result. Barometers, on the other hand, are likely to read too low instead of too high, o\nng to the imperfection of the vacuum and the action of capillary attractioa If the mercury be perfectly pure and no appreciable enor be due to the measuring apparatus, the best barometer will be that which gives the highest result. In determining the specific gravity of a solid body the chief danger of error arises from bubbles of air adhering to the body, which would tend to make the specific gravity too small Much attention must always be given to one-sided errors of this kind, since the multi- plication of experiments does not remove the error. In such cases one very careful experiment is better than any number of careless ones. When we have reasonable grounds for supposing that certain experimental results are liable to grave errors, we should exclude them in drawing a mean. If we want to find the most probable approximation to the velocity of ' Quetelet, Littert, &e. p. 1 16. xvilJ THE LAW 01*' EltliOK. 391 sound in air, it would be absurd to go back to the old experiments which made the velocity from 1200 to 1474 feet per second ; for we know that the old observers did not guard against errors arising from wind and other causes. Old chemical experiments are valueless as re- gards quantitative results. The old chemists found the atmospliere in different places to differ in composition nearly ten per cent., whereas modern accurate experi- menters find very slight variations. Any method of measurement which we know to avoid a source of error is far to be preferred to others which trust to probabilities for the elimination of the eiTor. As Flamsteed says,^ " One good instrument is of as much worth as a hundred in- different ones." But an instrument is good or bad only in a comparative sense, and no instrument gives invariable and truthful results. Hence we must always ultimately fall back upon probabilities for the selection of the final mean, when other precautions are exhausted. Legendre, the discoverer of the method of Least Squares, recommended that observations differing very much from the results of his method should be rejected. The subject has been carefully investigated by Professor Pierce, who has proposed a criterion for the rejection of doubtful observa- tions based on the following principle:' — observations should be rejected when the probability of the system of errors obtained by retaining them is less than that of the system of errors obtained by their rejection multiplied by the probability of making so many and no more abnormal observations." Professor Pierce's investigation is given nearly in his own words in Professor W. Chauvenet's '* Manual of Spherical and Practical Astronomy," which contains a full and excellent discussion of the methods of treating numerical observations.^ Very difficult questions sometimes arise when one or more results of a method of experiment diverge widely from the mean of the rest. Are we or are we not to ex- clude them in adopting the supposed true mean result of the method? The drawing of a mean result rests, as I ' Bailv, Account of Flamsiudy p. 56. ' Gould's Atironomical Journal^ Cambridffe, Mass., vol. ii. p. 161. • Philadelphia (London, Triibner) 1863. Appendix, vol. ii. p. 558. "ji 1 t I ! 392 THE PUINCIPLES OF SCIENCE. [chap i have frequently explained, upon the assumption that eveiy eiTor acting in one direction will probably be balanced by other errore acting in an opposite direction. If then we know or can possibly discover any causes of error not agreeing with this assumption, we shall be justified in excluding results which seem to be affected by this cause. In reducing large series of astronomical observations, it is not uncommon to meet with numbers diflfering from othei*s by a whole degree or half a degree, or some considerable in- tegral quantity. These are errors which could hardly arise in the act of observation or in instrumental irregularity ; but they might readily be accounted for by misreading of figures or mistaking of division marks. It would be absurd to trust to chance that such mistakes would balance each other in the long run, and it is therefore better to correct arbitrarily the supposed mistake, or better still, if new observations can be made, to strike out the diver- gent numbers altogether. When results come sometimes too great or too small in a regular manner, we should suspect that some part of the instrument slips through a definite space, or that a definite cause of error enters at times, and not at others. We should then make it a point of prime importance to discover the exact nature and amount of such an error, and either prevent its occurrence for the future or else introduce a corresponding correction. In many researches the whole difficulty will consist in this detection and avoidance of sources of error. Professor Hoscoe found that the presence of phosphorus caused serious and almost unavoidable enors in the determination of the atomic weight of vanadium.^ Herschel, in reducing his observations of double stars at the Cape of Good Hope,* was perplexed by an unaccountable difference of the angles of position as measured by the seven-feet equatorial and the twenty-feet reflector telescopes, and after a careful in- vestigation was obliged to be contented with introducing a correction experimentally determined.^ When observations are sufficiently numerous it seems desirable to project the apparent errors into a curve, and then to observe whether this curve exhibits the symmet- 1 Bakeriaii Lecture, PhiloMphical Trantactions (1868), vol. clviiL p. 6. ' Results of Observati&M ai the Cape of Oood Hope, p. 283. H XVIl.J THE LAW OF ERROR. 393 rical and characteristic form of the curve of error. If so, it may be inferred that the errors arise from many minute independent sources, and probably compensate each other in the mean result. Any considerable irregularity will indicate the existence of one-sided or large causes of error, which should be made the subject of investigation. Even the most patient and exhaustive investigations will sometimes fail to disclose any reason why some results diverge from others. The question again recurs — Are we arbitrarily to exclude them ? The answer should be in the negative as a general rule. The mere fact of divergence ought not to be taken as conclusive against a result, and the exertion of arbitrary choice would open the way to the fatal influence of bias, and what is com- monly known as the "cooking" of figures. It would amount to judging fact by theory instead of theory by fact. The apparently divergent number may prove in time to be the true one. It may be an exception of that valuable kind which upsets our false theories, a real exception, exploding apparent coincidences, and opening a way to a new view of the subject To establish this position for the divergent fact will require additional research ; but in the meantime we should give it some weight in our mean conclusions, and should bear in mind the discrepancy as one demanding attention. To neglect a divergent result is to neglect the possible clue to a great discovery. Method of Least Squares. When two or more unknown quantities are so involved that they cannot be separately determined by the Simple Method of Means, we can yet obtain their most probable values by the Method of Least Squares, without more difficulty than arises from the length of the arithmetical computations. If the result of each observation gives an equation between two unknown quantities of the form ax -\- by =■ c then, if the observations were free from error, we should need only two observations giving two equations; but for the attainment of greater accuracy, we may take many ol)- seryations, and reduce the equations so as to give only a pair with mean coefficients. This reduction is effected by dV4 TUE PRINCIPLES OF SCIENCE. [chap. JLTll.] THE LAW OF ERROR oSfD 1^ i / (i.), multiplying the coefficients of eacli equation by the first coefficient, and adding togetlier all the similar co- efficients thus resulting for the coefficients of a new equation ; and (2.), by repeating this process, and multi- plying tlie coefficients of each equation by the coeflicient of the second term. Meaning by (sum of a*) the sum of all quantities of the same kind, and having the same place in the equations as a-, we may briefly describe the two resulting mean equations as follows : — (sum of a*) . aj + (sum of ad) . y = (sum of ac), (sum of ab) . X -\- (sum of i«) . y = (sum of be). When there are three or more unknown quantities the process is exactly the same in nature, and we get additional mean equations by multiplying by the third, fourth, &c., coefficients. As the numbers are in any case approximate, it is usually unnecessary to make the com- putations with accuracy, and places of decimals may be freely cut off to save arithmetical work. The mean equations having been computed, their solution by the ordinary methods of algebra gives the most probable values of the unknown quantities. Works upon the Theory of ProbabUUy, Regarding the Theory of Probability and the Law of Error a s most important subjects of s ^^dy fnr any onft who desi resHx) obtain a compl e te compre benaioQ of scientific me tEod as actually ^applied in physical investigations, I will briefly indicafc^tne works in one or other of which the reader ^m^best "pursue the study. The best popular, and at the same time profound English work on the subject is De Morgan's "Essay on Proba- bilities and on their Application to Life Contingencies and Insurance Offices," published in the Gahirui Cyclopctdia, and to be obtained (in print) from Messrs. Longman. Mr. Venn's work on The Logic of Chance can now be procured in a greatly enlarged second edition ; * it contains a moat interesting and able discussion of the metaphysical ^ Thi Logic of Chance, an Essay on the FoundationB and Province of the Theory of Probability, with especial reference to its Logical Bearings and its Application to Moral and Social Science. (Mae> niiUau), 1876. basis of probability and of related questions concerning causation, belief, design, testimony, &c. ; but I cannot always agree with Mr. Venn's opinions. No mathematical knowledge beyond that of common arithmetic is required in reading these works. Quetelet's Letters form a good introduction to the subject, and the mathematical notes are of value. Sir George Airy's brief treatise On tlve Algebraical and Numerical Theory of Errors of Observa- tions and the Combination of Observations^ contains a complete explanation of the Law of Error and its prac- tical applications. De Morgan's treatise " On the Theory of Probabilities" in the Encyclopa^ia Metropolitana, presents an abstract oi the more abstruse investigations of Laplace, together with a multitude of profound and original remarks concerning the theory generally. In Lubbock and Drinkwater's work on Probability, in the Library of Useful Knowledge, we have a concise but good statement of a number of important problems. The Rev. W. A. Whitworth has given, in a work entitled Choice and Chance, a number of good illustrations of calculations both in combinations and probabilities. In Mr. Todhunter's admirable History we have an exhaustive critical account of almost all writings upon the subject of probability down to the culmination of the theory in I^place's works. The Memoir of Mr. J. W. L. Glaisher has already been mentioned (p. 375). In spite of the existence of these and some other good English works, there seems to be a want of an easy and yet pretty com- plete mathematical introduction to the study of the theory. Among French works the TraitS Elim^ntaire du Calcul des ProbabilUes, by S. F. Lacroix, of which several editions have been published, and which is not difficult to obtain, forms probably the best elementary treatise. Poisson's liecherches sur la FrobabilitS des Jugem&nts (Paris 1837), commence with an admirable investigation of the grounds and methods of the theory. While iSiplace's great Theorie Analytique des Frobabilitds is of course the " Principia " of the subject ; his Essai PhUosophiqu^e sur les Probability is a popular discourse, and is one of the most profound and interesting essays ever published. It should bo familiar to every student of logical method, and has lost little or none of its importance by lapse of time. 1 ID 1 i' 396 THE PRINCIPLES OF SCIENCE. [OBAP. Beleclion of Constant Errm's. The Method of Means is absolutely incapable of elimi- nating any error which is always the same, or which always lies in one direction. We sometimes require to be roused from a false feeling of security, and to be ui-ged to take suitable precautions against sucli occult errors. " It is to the observer," says Gauss,^ " that belongs the task of carefully removing the causes of constant errors," and this is quite true when the error is absolutely constant. When we have made a number of determinations with a certain apparatus or method of measurement, there is a great advantage in altering the arrangement, or even devising some entirely different method of getting estimates of the same quantity. The reason obviously consists in the im- probability that the same error will affect two or more different methods of experiment. If a discrepancy is found to exist, we shall at least be aware of the existence of error, and can take measures for finding in which way it lies. If we can try a considerable number of methods, the probabiUty becomes great that errors constant in one method will be balanced or nearly so by errors of an op- posite effect in the others. Suppose that there be thi-ee different methods each affected by an error of equal amount. The probability that this error will in all fall in the same direction is only J ; and with four methods similarly J. If each method be affected, as is always the case, by several independent sources of error, the probability becomes much gi-eater that in the mean result of all the methods some of the errora will partially compensate the others. In this case as in all others, when human vigilance has exhausted itself, we must trust the theory of probability. In the determination of a zero point, of the magnitude of the fundamental standards of time and space, in the personal equation of an astronomical observer, we have instances of fixed errors ; but as a general rule a change of procedure is likely to reverse the character of the error, and many instances may be given of the value of this precaution. If we measure over and over again the same ^ Qauas, translated by Beitiuud, p. a(. XVII.] THE LAW OF ERROR. 397 angular magnitude by the same divided circle, maintained in exactly the same position, it is evident that the same mark in the circle will be the criterion in each case, and any error in the position of that mark will equally affect all our results. But if in each measurement we use a different part of the circle, a new mark will come into use, and as the error of each mark cannot be in the same direction, the average result will be nearly free from errors of division. It will be better still to use more than one divided circle. Even when we have no perception of the points at which error is likely to enter, we may with advantage vary the construction of our apparatus in the hope that we shall accidentally detect some latent cause of error. Baily's purpose in repeating the experiments of Michell and Caven- dish on the density of the earth was not merely to follow the same course and verify the previous numbers, but to try whether variations in the size and substance of the attracting balls, the mode of suspension, the temperature of the surrounding air, &c., would yield different results. He performed no less than 62 distinct series, comprising 2153 experiments, and he carefully classified and discussed the results so as to disclose the utmost differences. Again, in experimenting upon the resistance of the air to the motion of a pendulum, Baily employed no less than 80 pendulums of various forms and materials, in order to ascertain exactly upon what conditions the resistance depends. Regnault, in his exact researches upon the dilatation of gases, made arbitrary changes in the magni- tude of parts of his apparatus. He thinks that if, in spite of such modification, the results are unchanged, the errors are probably of inconsiderable amount ; ^ but in reality it is always possible, and usually likely, that we overlook sources of error which a future generation will detect. Thus the pendulum experiments of Baily and Sabine were directed to ascertain the nature and amount of a correction for air resistance, which had been entirely misunderstood in the experiments by means of the seconds pendulum, upon which was founded the definition of the standard yard, in the Act of 5th George IV. c. 74. It has already * Jamin, Couts de Physique, vol. ii. p. 60. > 398 fHE PRINCIPLES OF SCIENCE. [ch. xtb. been mentioned that a considerable error was discovered in the detennination of the standard metre as the ten- millionth part of the distance from the pole to the equator (p. 3145. We shall return in Chapter XXV. to the further consi- deration of the methods by which we may as far as possible secure ourselves against permanent and undetected sources of error. In the meantime, having completed the con- sideration of the special methods requisite for treating quantitative phenomena, we must pursue our principal subject, and endeavour to trace out the course by which the physicist, from observation and experiment, collects the materials of knowledge, and then proceeds by hypo- thesis and inverse calcmation to induce fix)m them the laws of nature. BOOK III. INDUCTIVE INVESTIGATION. CHAPTER XVIII. OBSERVATION. All knowledge proceeds originally from experience. Using J the name in a wide sense, we may say that experience comprehends all that we ftd, externally or internally^ the aggregate of the impressions which we receive through the various apertures of perception — the aggregate con- sequently of what is in the mind, except so far as some portions of knowledge may be the reasoned equivalents of other portions. As the word experience expresses, we go throtigh much in life, and the impressions gathered inten- tionally or unintentionally afford the materials from which the active powers of the mind evolve science. No small part of the experience actually employed in science is acquired without any distinct purpose. We cannot use the eye:^ without gathering some facts which may prove useful. A great science has in many cases risen from an accidental observation. Erasmus Bartholinus thus first discovered double refraction in Iceland spar; Galvani noticed the twitching of a frog's leg; Oken was struck by the form of a vertebra; Mains accidentally examined light reflected from distant windows with a It ' '1 400 THE PRINCIPLES OP SCIENCE. [chap. M, i ) r> \ double refracting substance ; and Sir John Hei-schel's attention was drawn to the peculiar appearance of a solution of quinine sulphate. In earlier times there must have been some one who first noticed the strange behaviour of a loadstone, or the unaccountable motions produced by amber. As a general rule we shall not know in what direction to look for a great body of phenomena widely different from those familiar to us. Chance then must give us the starting point ; but one accidental observation well used may lead us to make thousands of observations in an intentional and organised manner, and thus a science may be gradually worked out from the smallest opening. Distinction of Observation and Sxperi/runt. It is usual to say that the two sources of experience are Observation and Experiment. When we merely note and record the phenomena which occur around us in the ^Ci)rdinary course of nature we are said to observe. When we change the course of nature by the intervention of our nmscular powers, and thus produce unusual combinations and conditions of phenomena, we are said to experiment. Herschel justly remarked ^ that we might properly call these two modes of experience passive and active observa- tion. In both cases we must certainly employ our senses to observe, and an experiment difiers from a mere observar tion in the fact that we more or less influence the character of the events which we observe. Experiment is ^ thus observation plus alteration of conditions. It may readily be seen that we pass upwards by in- sensible gradations from pure observation to determinate experiment When the earliest astronomers simply noticed the ordinary motions of the sun, moon, and planets upon the face of the starry heavens, they were pure obseiTers. <^ But astronomers now select precise times and places for important observations of stellar parallax, or the transits of planets. They make the earth's orbit the basis of a .well arranged natural experivient, as it were, and take well considered advantage of motions which they cannot controL Meteorology might seem to be a science of pure ' Preliminary Difcourse on the Study of Naimral Pkiloiophy, p. 77. xviil] OBSERVATION. 401 observation, because we cannot possibly govern the changes of weather which we record. Nevertheless we may ascend mountains or rise in balloons, like Gay-Lussac and Glaisher, and may thus so vary the points of observation as to render our procedure experimental. We are wholly unable either to produce or prevent earth-currents of electricity, but when we construct long lines of telegraph, we gather such strong currents during periods of disturbance as to render them capable of easy observation. The best arranged systems of observation, however, would fail to give us a large part of the facts which we now possess. Many processes continually going on in nature are so slow and gentle as to escape our powers of observa- tion. Lavoisier remarked that the decomposition of water must have been constantly proceeding in nature, although its possibility was unknown till his time.^ No substance is wholly destitute of magnetic or diamagnetic powers ; but it required all the experimental skill of Faraday to prove that iron and a few other metals had no monopoly ^.^^ of these powers. Accidental observation long ago im-^y pressed upon men's minds the phenomena of lightning, and the attractive properties of amber. Experiment only could have shown that phenomena so diverse in magnitude and character were manifestations of the same agent. To observe with accuracy and convenience we must have agents under our control, so as to raise or lower their intensity, to stop or set diem in action at will. Just as Smeaton found it requisite to create an artificial and governable supply of wind for his investigation of wind- mills, so we must have governable supplies of light, heat, electricity, muscular force, or whatever other agents we are examining. It is hardly needful to point out too that on the earth's surface we live under nearly constant conditions of gravity, temperature, and atmospheric pressure, so that if we are to extend our inferences to other parts of the universe where conditions are widely diflTerent, we must be prepared to imitate those conditions on a small scale here. We must have intensely high and low temperatures ; we must vary 111 < • ;. * Layoisier's EUmmtt of Chemistry , translated by Kerr, 3rd ed. p. 148. n D 402 THE PRINCIPLBS OP SCIENCE. [chap m the density of gases from approximate vacuum upwards ; we must subject liquids aud solids to pressures or stmins of almost unlimited amount. Mental Conditions of Correct Observation, Every observation must in a cei-tain sense be true, foi the observing and recording of an event is in itself an event But before we proceed to deal with the supposed meaning of the record, and dmw inferences concerning the course of nature, we must take care to ascertain that the character and feelings of the observer are not to a great extent the phenomena recorded. The mind of man, as Francis Bacon said, is like an uneven mirror, and does not reflect the events of nature without distortion. We need hardly take notice of intentionally false observations, nor of mistakes arising from defective memory, deficient light, and so forth. Even whei*e the utmost fidelity and care are used in observing and recoixling, tendencies to error exist, and fallacious opinions arise in consequence. It is difficult to find persons who can with perfect fair- ness register facts for and against their own peculiar views. Among uncultivated observers the tendency to remark favourable and foi-get unfavourable events is so great, that no reliance can be placed upon their supposed observations. Thus arises the enduring fallacy that the changes of the weather coincide in some way with the changes of the moon, although exact and impartial registei*s give no countenance to the fact. The whole race of prophets and quacks live on the overwhelming eftfect of one success, compared with hundreds of failures wliich are unmen- tioned and forgotten. As Bacon says, ** Men mark when they hit, and never mark when they miss." And we should do well to bear in mind the ancient story, quoted by Bacon, of one who in Pagan times was shown a temple with a picture of all the persons who had been saved from shipwreck, after paying their vows. When asked whether he did not now acknowledge the power of the gods, "Ay," he answered; "but where are they painted that were drowned after their vows ? " If indeed we could estimate the amount of bias existing in any particular observations, it might be treated like XVIII.] OBSERVATION. 409 one of the forces of the problem, and the true course of external nature might still be rendered apparent. But the feelings of an observer are usually too indeterminate, so ^ that when there is reason to suspect considerable bias, re- jection is the only safe course. As regards facts casually registered in past times, the capacity and impartiality of the observer are so little known that we shoiid spare no pains to replace these statements by a new appeal to nature. An indiscriminate medley of truth and absurdity, such as Francis Bacon collected in his Natural History, is wholly unsuited to the purposes of science. But of course when records relate to past events like eclipses, con- junctions, meteoric phenomena, earthquakes, volcanic eruptions, changes of sea margins, the existence of now extinct animals, the migrations of tribes, remarkable customs, &c., we must make use of statements however unsatisfactory, and must endeavour to verifiy them by the comparison of independent records or traditions. When extensive series of observations have to be made, as in astronomical, meteorological, or magnetical observa- tories, trigonometrical surveys, and extensive chemical or physical researches, it is an advantage that the numerical work should be executed by assistants who are not interested ^ in, and are perhaps unaware of, the expected results. The record is thus rendered perfectly impartial. It may even be desirable that those who perform the purely routine work of measurement and computation should be un- acquainted with the principles of the subject. The great table of logarithms of the French Revolutionary (jovem- ment was worked out by a staff of sixty or eighty computers, most of whom were acquainted only with the rules of arithmetic, and worked under the direction of skilled mathematicians ; yet their calculations were usually found more correct than those of persons more deeply t/ versed in mathematics.^ In the Indian Ordnance Survey the actual measurers were selected so that they should not have sufficient skill to falsify their results without detection. Both passive observation and experimentation must, however, be generally conducted by persons who know for ^ Babbage, Economy nf Many/ticturetf p. 194. D D 2 Hit 404 THE PRINCIPLES OF SCIENCE. [chap. what they are to look. It is only when excited and guided by the hope of verifying a theory that the observer will notice many of the most important points ; and, where the work is not of a routine character, no assistant can super- sede the mind-direct-ed observations of the philosopher. Thus the successful investigator must combine diverse qualities ; he must have clear notions of the result he ex- pects and confidence in the truth of his theories, and yet he must have that candour and flexibility of mind which enable him to accept unfavourable results and abandon mistaken views. Instrumental and Sensual Condttums of Observation. In every observation one or more of the senses must be employed, and we should ever bear in mind that the ex- tent of our knowledge may be limited by the power of the sense concerned. What we learn of the world only forms the lower limit of what is to be learned, and, for all that we can tell, the processes of nature may infinitely sur- pass in variety and complexity those which are capable of coming within our means of observation. In some cases inference from observed phenomena may make us in- directly aware of what cannot be directly felt, but we can never be sure that we thus acquire any appreciable fraction of the knowledge that might be acquired. It is a strange reflection that space may be filled with dark wandering stars, whose existence could not have yet become in any way known to us. The planets have already cooled so far as to be no longer luminous, and it may well be that other stellar bodies of various size have fallen into the same condition. From the consideration, indeed, of variable and extinguished stars, Laplace inferred that there probably exist opaque bodies as great and perhaps as numerous as those we see.^ Some of these dark stars might ultimately become known to us, either by reflecting light, or more probably by their gravitating effects upon luminous stars. Thus if one member of a Jouble star were dark, we could readily detect its exist- ence, and even estimate its size, position, and motions. ' System of the Worlds tranalated by Harte, vol ii. p. 335 ZVIll.] OBSERVATION. 403 by observing those of its visible companion. It was a favourite notion of Huygher.s that there may exist stars and vast universes so distant that their light has never yet had time to reach our eyes ; and we must also bear in mind that light may possibly suffer slow extinction in space, so that there is more than one way in which an absolute limit to the powers of telescopic discovery may exist There are natural limits again to the power of our senses in detecting undulations of various kinds. It is commonly said that vibrations of more than 38,000 strokes per second are not audible as sound ; and as some eare actually do hear sounds of much higher pitch, even two octaves higher than what other ears can detect, it is exceedingly probable that there are incessant vibnitions which we cannot call sound because they are never heard. Insects may communicate by such acute sounds, con- stituting a language inaudible to us ; and the remarkable agreement apparent among bodies of ants or bees might thus perhaps be explained. Nay, as Fontenelle long ago suggested in his scientific romance, there may exist un- limited numbers of senses or modes of perception which we can never feel, though Darwin's theory woidd render it probable that any useful means of knowledge in an an- cestor would be developed and improved in the descendants. We might doubtless have been endowed with a sense capable of feeling electric phenomena with acuteness, so that the positive or negative state of charge of a body could be at once estimated. The absence of such a sense is probably due to its comparative uselessness. Heat undulations are subject to the same considerations. It is now apparent that what we call light is the affection of the eye by certain vibrations, the less rapid of which are invisible and constitute the dark rays of radiant heat, in detecting which we must substitute the thermometer pr the thermopile for the eye. At the other end of the spectnim, again, the ultra-violet rays are invisible, and only indirectly brought to our knowledge in the pheno- mena of fluorescence or photo-chemical action. There is no reason to believe that at either end of the spectrum an absolute limit has yet been reached. Just as our knowledge of the stellar univei-se is limited "ill : t (^ 11 . m i !! :| y ^ \ 406 THE PRINCIPLES OF SCIENCE. [OHAf. by the power of the telescope and other conditions, so our knowledge of the minute world has its limit in the powers and optical conditions of the microscope. There was a time when it would have been a reasonable induction that vegetables are motionless, and animals alone endowed with power of locomotion. We are astonished to dis- cover by the microscope that minute plants are if any- thing more active than minute animals. We even find that mineral substances seem to lose their inactive character and dance about with incessant motion when reduced to suflRciently minute particles, at least when sus- pended in a non-conducting medium.^ Microscopists will meet a natural limit to observation when the minuteness of the objects examined becomes comparable to the length of light undulations, and the exti'eme difficulty already encountered in determining the forms of minute marks on Diatoms appears to be due to this cause. According to Helmholtz the smallest distance which can be accurately defined depends upon the interference of light passing through the centres of the bright spaces. With a the- oretically perfect micrascope and a diy lense the smallest visible object would not be less than one 8o,ocx)th part of an inch in red light. Of the errors likely to arise in estimating quantities by the senses I have already spoken, but there are some cases in which we actually see things diflFerently from what they are. A jet of water appears to be a continuous thread, when it is really a wonderfully organised succes- sion of small and large drops, oscillating in form. The drops fall so rapidly that their impressions upon the eye i-un into each other, and in order to see the separate drops we require some device for giving an instantaneous view. One insuperable limit to our powers of observation arises from the impossibility of following and identifying the ultimate atoms of matter. One atom of oxygen is probably undistinguishable from another atom; only by ^ Tliis curions plicuomcnon, which I propose to call pedetis, or the pedctic movanentf from ttjSow, to jump, is carefnlly described in my paper published in the Quarterly Journal of Science for April, 1878, vol. riii. (N.S.) p. 167. See aI«o Proceedings of the Literary and Philo$cpMcal Society tf Manchester, 2Sth Janiuiry, 1870, vol. ix. p. 78, Nature^ 22nd August, 1878, vol. xTiii. p 44(\ or the QiMrUrly Journal of Science, vol. riii. tN.8.)p. 514. XVIII.] OBSERVATION. 407 keeping a certain volume of oxygen safely inclosed in a bottle can we assure ourselves of its identity ; allow it to mix with other oxygen, and we lose all power of iden- tification. Accordingly we seem to have no means of directly proving that every gas is in a constant state of diffusion of every part into every part. We can only infer this to be the case from observing the behaviour of distinct gases which we can distinguish in their course, and by reasoning on the grounds of molecular theory.^ External Conditions of Correct Observation, fy Before we proceed to draw inferences from any series of recorded facts, we must take care to ascertain perfectly, if possible, the external conditions under which the facts are brought to our notice. Not only may the observing mind be prejudiced and the senses defective, but there may be circumstances which cause one kind of event to come more frequently to our notice than another. The comparative numbers of objects of different kinds existing may in any degree differ from the numbers which come to our notice. This difference must if possible be taken into account before we make any inferences. There long appeared to be a strong presumption that all comets moved in elliptic orbits, because no comet had been proved to move in any other kind of path. The theory of gravitation admitted of the existence of comets moving in hyperbolic orbits, and the question arose whether they were really non-existent or were only beyond the bounds of easy observation. From reason- able suppositions Laplace calculated that the probability was at least 6000 to i against a comet which comes within the planetary system sufficiently to be visible at the earth's surface, presenting an orbit which could be discriminated from a very elongated ellipse or parabola in the i>art of its orbit within the reach of our telescopes.^ In short, the chances are very much in favour of our seeing elliptic rather than hyperbolic comets. Laplace's views have been confirmed by the discovery of six * Maxwell, Theory of Heat, p. 301. « Ijaplace, EsMoi FhilosopkiquCf p. 59. Todhunter's Uiitory, pp. 491—494. T ■:L-',^-.-J.^M.,f.^J^ i] I II I L In I' I i 1 1 408 THE PRINCIPLES OF SCIENCE. [chap. hyperbolic comets, which appeared in the years 1729, 1 77 1, 1774, 18 18, 1840, and 1843,^ and as only about 800 comets altogether have been recorded, the proportion of hyperbolic ones is quite as large as should be expected. When we attempt to estimate the numbers of objects which may have existed, we must make large allowances for the limited sphere of our observations. Probably not more than 4000 or 5000 comets have been seen in historical times, but making allowance for the absence of observers in the southern hemisphere, and for the small probability that we see any considerable fraction of those which are in the neighbourhood of our system, we must ^ accept Kepler's opinion, that there are more comets in the regions of space than fishes in the depths of the ocean. When like calculations are made concerning the numbers of meteors visible to us, it is astonishing to find that the number of meteors entering the earth's atmosphere in every twenty-four hours is probably not less than 400,000,000, of which 13,000 exist in every portion of space equal to that filled by the earth. Serious fsdlacies may arise from overlooking the inevit- able conditions under which the records of past events arc brought to our notice. Thus it is only the durable objects manufactured by former races of men, such as flint imple- ments, which can have come to our notice as a general rule. The comparative abundance of iron and bronze articles used by an ancient nation must not be supposed to be coincident with their comparative abundance in our museums, because bronze is far Uie more durable. There is a prevailing fallacy that our ancestors built more strongly than we do, arising from the fact that the more fragile structures have long since crumbled away. We have few or no relics of the habitations of the poorer classes among the Greeks or Eomaus, or in fact of any past race ; for the temples, tombs, public buildings, and mansions of the wealthier classes alone endure. There is an immense expanse of past events necessaiily lost to us for ever, and we must generaUy look upon records or relics as exceptional in their character. The same gonsiderations apply to geological relics. We could not generally expect that animals would be ^ Ciuuubexs' Attronomfff ut ed. |^ 203. ^ XVI II.] OBSERVATION. 4(19 preserved unless as regaixis the bone^, shells, strong integu- ments, or other hard and durable parts. All the infusoria and animals devoid of mineral framework have probably perished entirely, distilled perhaps into oils. It has been pointed out tliat the peculiar character of some extinct floras may be due to the unequal preservation of different families of plants. By various accidents, however, we gain glimpses of a world that is usually lost to us — as by insects embedded in amber, the great mammoth preserved in ice, mummies, casts in solid material like that of the Koman soldier at Pompeii, and so forth. We should also remember, that just as there may be conjunctions of the heavenly bodies that can have hap- pened only once or twice in the period of history, so re- markable terrestrial conjunctions may take place. Great storms, earthquakes, volcanic eruptions, landslips, floods, irruptions of the sea, may, or rather must, have occurred, events of such unusual magnitude and such extreme rarity that we can neither expect to witness them nor readily to comprehend their effects. It is a great advantage of the study of probabilities, as Laplace himself remarked, to make us mistrust the extent of our knowledge, and pay proper regard to the probability that events would come within the sphere of our observations. Appareiit Sequence of Events, De Morgan has excellently pointed out^ that there are no less than four modes in which one event may seem to follow or be connected with another, without being really so. These involve mental, sensual, and ex- ternal causes of error, and I will briefly state and illustrate them. Instead of A causing B, it may be our perception of A that causes B, Thus it is that prophecies, presentiments, and the devices of sorcery and witchcraft often work their own ends. A man dies on the day which he has always regaixled as his last, from his own fears of the day. An incantation effects its purpose, because care is taken to frighten the intended victim, by letting him know his fate. In all such cases the mental condition is the caus^ of apparent coincidence. * JSffay on Probabilitiet, Cabinet Cyclopaedia, p. 121. n\ li [ / (/ ■ 410 THE PRINCIPLES OF SCIENCE. [OHAR } In a second class of cases, the event A may make our perception of B follow, which waidd otherwise happen withmU being perceived. Thus it was believed to be the result of investigation that more comets appeared in hot than cold summers. No account was taken of the fact that hot summers would be comparatively cloudless, and afford better opportunities for the discovery of comets. Here the disturbing condition is of a purely external character. Certain ancient philosophers held that the moon's rays were cold-producing, mistaking the cold caused by radiation into space for an effect of the moon, which is more likely to be visible at a time when the absence of clouds permits radiation to proceed. In a third class of cases, our perception of A may makt our perception of B follow. The event B may be con- stantly happening, but our attention may not be drawn to it except by our observing A. This case seems to be illustrated by the fallacy of the moon's influence on clouds. The origin of this fallacy is somewhat complicated. In the first place, when the sky is densely clouded the moon would not be visible at all ; it would be necessary for us to see the full moon in order that our attention should be strongly drawn to the fact, and this would happen most often on those nights when the sky is cloudless. Mr. W. Ellis,^ moreover, has ingeniously pointed out that there is a general tendency for clouds to disperse at the com- mencement of night, which is the time when the full moon rises. Thus the change of the sky and the rise of the full moon are likely to attract attention mutually, and the coincidence in time suggests the relation of cause and effect. Mr. Ellis proves from the results of observations at the Greenwich Observatory that the moon possesses no appreciable power of the kind supposed, and yet it is remarkable that so sound an observer as Sir John Herschel was convinced of the connection. In his " Results of Observations at the Cape of Good Hope,"* he mentions many evenings when a full moon occurred with a pecidiarly clear sky. > Philoiophical Magazine^ 4th Series (1867), toI. xxxiv. p. 64. * See NoUi to Measures of Double StarSy 1204, 1336, 1477, 1686, 1786, 1816, 1835, 1929, 2081, 2186, pp. 265, &c. See also Herschers Familiar Lectures on Scientific SubjectSf p 147, and Outlines of Astronomy f 7th ed. p. 28c xviii.] OBSERVATION. 411 V^ There is yet a fourth class of cases, in which B is really the arUecedevt event, but our perception of A, which is a consequence of B, may he necessary to bring about our perception of B. There can be no doubt, for ^instance, that upward and downward currents are continually cir- culating in the lowest stratum of the atmosphere during the day-time ; but owing to the transparency of the at- mosphere we have no evidence of their existence until we perceive cumulous clouds, which are the consequence of such currents. In like manner an interfiltration of bodies of air in the higher parts of the atmosphere is probably in nearly constant progress, but unless threads of cirrous cloud indicate these motions we remain ignorant of their occurrence.' The highest strata of the atmosphere are wholly imperceptible to us, except when rendered luminous by auroral currents of electricity, or by the passage of meteoric stones. Most of the visible phenomena of comets probably arise from some substance which, existing pre- viously invisible, becomes condensed or electrified suddenly into a visible form. Sir John Herschel attempted ti) explain the production of comet tails in this manner by evaporation and condensation.* Negative Arguments from Non-ohservation. Fiom what has been suggested in preceding sections, it will plainly appear that the non-observation of a pheno- menon is not generally to be taken as proving its non- occurrence. As there are sounds which we cannot hear, rays of heat which we cannot feel, multitudes of worlds which we cannot see, and myriads of minute organisms of which not the most powerful microscope can give us a view, we must as a general rule interpret our experience in an affirmative sense only. Accordingly when inferences have been drawn from the non-occurrence of particular facts or objects, more extended and careful examination has often proved their falsity. Not many years since it was quite a well credited conclusion in geology that no remains of man were found in connection with those of * Jevons, On the Cxrrous Form of Cloud, Philosophical Magazine, July, 1857, 4th Series, vol- xiv. p. 22. ' Astronomy, 4th ed. p. 358 'H ilS THE PRINCIPLES OP SCIENCE. IcuaP. XTIfl.T OBSERVATION. 418 I V / o extinct animals, or in any deposit not actually at present in course of formation. Even Babbage accepted this con- clusion as strongly confirmatory of the Mosaic accounts.' While the opinion was yet universally held, flint imple- ments had been found disproving such a conclusion, and overwhelming evidence of man's long-continued existence has since been forthcoming. At the end of the last century, when Herschel had searched the heavens with his powerful telescopes, there seemed little probability that planets yet remained unseen within the orbit of Jupiter. But on the first day of this century such an opinion was overturned by the discovery of Ceres, and more than a hundred other small planets have since been added to the lists of the planetary system. The discovery of the Eozoon Canadense in strata of much greater age than any previously known to contain organic remains, has given a shock to groundless opinions concerning the origin of organic forms; and the oceanic dredging expeditions under l3r. Carpenter and Sir Wy ville Thomson have modified some opinions of geologists by disclosing the continued existence of forms long supposed to be extinct. These and many other cases which might be quoted show the extremely unsafe character of negative inductions. But it must not be supposed that negative arguments are of no force and value. The earth's surface has been sufficiently searched to render it highly improbable that any terrestrial animals of the size of a camel remain to be discovered. It is believed that no new large animal has been encountered in the last eighteen or twenty centuries,* and the probability that if existent they would have been seen, increases the probability that they do not exist. We may with somewhat less confidence discredit the existence of any large unrecognised fish, or sed animals, such as the alleged sea-serpent. But, as we descend to forms of smaller size negative evidence loses weight from the less probability of our seeing smaller objects. Even the strong induction in favour of the four-fold division of the animal kingdom into Vertebrata, Annulosa, Mollusca, ' Babbage, Ninth Bridgetoater Treatise^ p. 67. ■ Cuvier, Esiay on the Theory 0/ Uu Earth, trauaUition, p. 61, &C. and Ccelenterata, may break down by the discovery of in- termediate or anomalous forms. As civilisation spreads over the surface of the earth, and unexplored tracts are gradually diminished, negative conclusions will increase in force ; but we have much to learn yet concerning the depths of the ocean, almost wholly unexamined as they are, and covering three-fourths of the earth's surface. In geology there are many statements to which con- siderable probability attaches on account of the large extent of the investigations already made, as, for instance, that true coal is found only in rocks of a particular geolo- gical epoch ; that gold occurs in secondary and tertiary strata only in exceedingly small quantities,^ probably derived from the disintegration of earlier rocks. In natural history negative conclusions are exceedingly treacherous and unsatisfactoiy. The utmost patience will not enable a microscopist or the observer of any living thing to watch the behaviour of the organism under all ciVcumstances continuously for a great length of time. There is always a chance therefore that the critical act or change may take place when the observer's eyes are with- drawn. This certainly happens in some cases ; for though the fertilisation of orchids by agency of insects is proved as well as any fact in natural history, Mr. Darwin has never been able by the closest watching to detect an insect in the performance of the operation. Mr. Darwin has himself adopted one conclusion on negative evidence, namely, that the Orchis pyramidalis and certain other orchidaceous flowers secrete no nectar. But his caution and unwearying patience in verifying the conclusion give an impressive lesson to the observer. For twenty-three consecutive days, as he tells us, he examined flowers in all states of the weather, at all hours, in various localities. As the secretion in other flowers sometimes takes place rapidly and might happen at early dawn, that inconvenient hour of observation was specially adopted. Flowers of different ages were subjected to irritating vapours, to mois- ture, and to every condition likely to bring on the secretion ; and only after invariable failure of this exhaustive inquir}' was the barrenness of the nectaries assumed to be proved' 1 Murchison's SUuriaf ist ed. p. 432. * Darwin's Fertiluation of Orehids, p. 48. I ■**C^i'. i !< I /I ^ t I ? 414 THE PRINCIPLES OP SCIENCE. [OHAf. In order that a negative argument founded on the non- observation of an object shall have any considerable force, it must be shown to be probable that the object if existent /<^ would have been observed, and it is this probability which defines the value of the negative conclusion. The failure of astronomers to see the planet Vulcan, supposed by some to exist within Mercury's orbit, is no sufficient disproof of its existence. Similarly it would be very difficult, or even impossible, to disprove the existence of a second satellite of small size revolving round the earth. But if any person ' make a particular assertion, assigning place and time, then oDservation will either prove or disprove the alleged fiict. If it is true that when a French observer professed to have seen a planet on the sun's face, an observer in Brazil was carefully scrutinising the sun and failed to see it, we have a negative proof. False facts in science, it has been well said, are more mischievous than false theories. A false theory is open to every person's criticism, and is ever liable to be judged by its accordance with facts. But a false or grossly erroneous assertion of a fact often stands in the way of science for a long time, because it may be extremely difficult or even impossible to prove the falsity of what has been once recorded. In other sciences the force of a negative argument will often depend upon the number of possible alternatives which may exist. It was long believed that the quality of a musical sound as distinguished from its piteh, must depend upon the form of the undulation, because no other cause of it had ever been suggested or was apparently possible. The truth of the conclusion was proved by Helmholtz, who applied a microscope to luminous points attached to the strings of various instruments, and thus actually observed the different modes of undulation. In mathematics negative inductive arguments have seldom much force, because the possible forms of expres- sion, or the possible combinations of lints and circles in geometry, are quite unlimited in number. An enormous / number of attempts were made to trisect the angle by the/ ordinary methods of Euclid's geometry, but their in/^ variable failure did not establish the impossibility of the task. This was shown in a totally different manner, by proving that the problem involves an inedudble cubic ^ X^lll.l OBSERVATION. 415 equation to which there could be geometrical solution.^ This is ahsurdum, a form of argument character. Similarly no number general solution of equations of establish the impossibility of the mode, equivalent to a reductio ad bility is considered to be proved. no corresponding plane a case of redicctio ad of a totally different of failures to obtain a the fifth degree would t^k, but in an indirect absurdum, the impossi- ^ Peacock, AlgAre, voL ii. p. 344. * Ibid, p. 359. Serret, Alg^bre SupSriewt^ and ed. p. 304.. ' s . / M CHAPTER XIX. BXPERIMENT. We may now consider the great advantages which we enjoy in examining the combinations of phenomena when things are within our reach and capable of being experi- mented on. We are said to experiment when we bring sub- stances together under various conditions of temperature, ^ pressure, electric disturbance, chemical action, &c., and then record the changes observed. Our object in induc- tive investigation is to ascertain exactly the group of cir- cumstances or conditions which being present, a certain other group of phenomena will follow. If we denote by A the antecedent group, and by X subsequent pheno- mena, our object will usually be to discover a law of the form A = AX, the meaning of which is that where A is X will happen. The circumstances which might be enumerated as present in the simplest experiment are very numerous, in fact al- most infinite. Rub two sticks together and consider what would be an exhaustive statement of the conditions. There are the form, hardness, organic sti-ucture, and all the chemical qualities of the wood; the pressure and velocity of the rubbing ; the temperature, pressure, and all the chemical qualities of the surrounding air ; the proxi- mity of the earth with its attractive and electric powers ; the temperature and other properties of the persons pro- ducing motion ; the radiation from the sun, and to and from the sky ; the electric excitement possibly existing in any overhanging cloud ; even the positions of the heavenly bodies must be mentioned. On d priori grounds it is CHAP. XIX.] EXPERIMENT. 417 unsafe to assume that any one of these circumstances is without effect, and it is only by experience that we can single out those precise conditions from which the observed ^^eat of friction proceeds. f J The great method of experiment consists in removing, ^one at a time, each of those conditions which may be imagined to have an influence on the result. Our object in the experiment of rubbing sticks is to discover the exact circumstances under which heat appears. Kow the pre- sence of air may be requisite ; therefore prepare a vacuum, and rub the sticks in every respect as before, except that it is done in vacuo. If heat still appears we may say that air is not, in the presence of the other circumstances, a requisite condition. The conduction of heat from neigh- bouring bodies may be a condition. Prevent this by mak- ing all the surrounding bodies ice cold, which is what Davy aimed at in rubbing two pieces of ice together. If heat still appears we have eliminated another condition, and so we may go on until it becomes apparent that the expen- diture of energy in the friction of two bodies is the sole condition of the production of heat. The great difficulty of experiment arises from the fact •// that we must not assume the conditions to be independent P revious to experime nt we have no right to say that the rubbing of two sticks will produce heat in the same way when air is absent as before. We may have heat produced in one way when air is present, and in another when air is absent The inquiry branches out into two lines, and we ought to try in both cases whether cutting off a supply of heat by conduction prevents its evolution in friction. The same branching out of the inquiry occurs with regard to every circumstance which enters into the experiment Regarding only four circumstances, say A, B, C, D, we ought to test not only the combinations ABCD, ABCrf, ABcD, A6CD, aBCD, but we ought really to go through the whole of the combinations given in the fifth column of the Logical Alphabet. The effect of the absence of each condition should be tried both in the presence and absence of every other condition, and every selection of those conditions. Perfect and exhaustive experimentation would, in short, consist in examining natural phenomena in all their possible combinations and registering al] K E 4td THE PRINCIPLES 0^ SCIENCE. [cHAf II ^ ft relations between conditions and results which are found capable of existence. It would thus resemble the exclusion of contradictory combinations carried out in the Indirect Method of Inference, except that the exclusion of com- binations is grounded not on prior logical premises, but on a posteriori results of actual trial. The reader will perceive, however, that such exhaustive investigation is practically impossible, because the number of requisite experiments would be immensely great Four antecedents only would require sixteen experiments; twelve antecedents would require 4096, and the number increases as the powers of two. The result is that the experimenter lias to fall back upon his own tact and experience in select- ing those experiments which are most likely to yield him significant facts. It is at this point that logical rules and forms begin to fail in giving aid. The logical rule is— Try all possible combinations; but this being impmcticable, the experimentalist necessarily abandons strict logical method, and trusts to his own insight. Analogy, as we shall see, gives some assistance, and attention sliould be concentrated on those kinds of conditions which have been found important in like cases. But we are now entirely in the region of probability, and the experimenter, while he is confidently pursuing what he thinks the right clue, may be overlooking the one condition of importance. It is an impressive lesson, for instance, that Newton pursued all his exquisite researches on the spectrum unsuspicious of the fact that if he reduced the hole in the shutter to a narrow slit, all the mysteries of the bright and dark lines were within his grasp, provided of course that his prisms were sufficiently good to define the rays. In like manner we know not what slight alteration in the most familiar experiments may not open the way to realms of new discovery. Practical difficulties, also, encumber the progi-ess of the physicist. It is often impossible to alter one condition without altering others at the same time; and thus we may not get the pure effect of the condition in question. Some conditions may be absolutely incapable of alteration ; others may be with great difficulty, or only in a certain degree, removable. A very treacherous source of error is the existence of unknown conditions, which of coujse we xul] EXPERIMENT. 419 cannot remove except by accident These difficulties we will shortly consider in succession. It is beautiful to observe how the alteration of a single circumstance sometimes conclusively explains a pheno- menon. An instance is found in Faraday's investigation of the behaviour of Lycopodium spores scattered on a vibrating plate. It was observed that these minute spores collected together at the points of greatest motion, whereas sand and all heavy particles collected at the nodes, where the motion was least It happily occurred to Faraday to try the experiment in the exhausted receiver of an air- pump, and it was then found that the light powder behaved exactly like heavy powder. A conclusive proof was thus obtained that the presence of air was the condition of im- portance, doubtless because it was thrown into eddies by the motion of the plate, and carried the Lycopodium to the points of greatest agitation. Sand was too heavy to be carried by the air. Exclusion of Indifferent Circumstances, From what has been already said it will be apparent that the detection and exclusion of indifferent circum- stances is a work of importance, because it allows the concentration of attention upon circumstances which con- tain the principal condition. Many beautiful instances may be given where all the most obvious antecedents have been shown to have no part in the production of a phenomenon. A pei-son might suppose that the peculiar colours of mother- of-pearl were due to the chemical qualities of the substance Much trouble might have been spent in following out that notion by comparing the chemical qualities of various iri- descent substances. But Brewster accidentally took an; impression from a piece of mother-of-pearl in a cement of resin and bees'-wax, and finding the colours repeated upon the surface of the wax, he proceeded to take other impres- sions in balsam, fusible metal, lead, gum arable, isinglass, &c., and always found the iridescent colours the same. He thus proved that the chemical nature of the substance is a matter of indifference, and that the form of the surface is the real condition of such colours.^ Nearly the same may " frtatiu on, Optiu^ by Brewster, Cab. Cyclo. p. 1 17. ? F 2 Hil lir it i» ll« i^ 480 THE PRINCIPLES OF SCIENCE. [OHAP. be said of the colours exhibited by thin plates and films. The rings and lines of colour will be nearly the same in character whatever may be the nature of the substance ; nay, a void space, such as a crack in glass, would produce them even though the air were withdrawn by an air-pump. The conditions are simply the existence of two reflecting surfaces separated by a very small space, though it should be added that the refractive index of the intervening sub- stance has some influence on the exact nature of the colour produced. When a ray of light passes close to the edge of an opaque body, a portion of tlie light appears to be bent towards it, and produces coloured fringes within the shadow of the body. Newton attributed this inflexion of light to the attraction of the opaque body for the supposed particles of liglit, although he was aware that the nature of the sur- rounding medium, whether air or other pellucid substance, exercised no apparent influence on the phenomena. Gravesande proved, however, that the character of the fringes is exactly the same, whether the body be dense or rare, compound or elementary. A wire produces exactly the same fringes as a hair of the same thickness. Even the form of the obstructing edge was subsequently shown to be a matter of indifference by Fresnel, and the interfer- ence spectrum, or the spectrum seen when light passes through a fine grating, is absolutely the same whatever be the form or chemical nature of the bars making the grating. Thus it appeara that the stoppage of a portion of a beam of light is the sole necessary condition for the diffraction or inflexion of light, and the phenomenon is shown to bear no analogy to the refraction of light, in which the form and nature of the substance are all impor- tant. It is interesting to observe how carefully Newton, in his researches on the spectrum, ascertained the indifference of many circumstances by actual trial. He says : * " Now the different magnitude of the hole in the window-shut, and different thickness of the prism where the rays passed through it, and different inclinations of the prism to the horizon, made no sensible changes in the length of the ■ Oflidu, 3rd. ed. p. 2$. I '1 {■i\ XIX.] EXPERIMENT. 421 image. Neither did the different matter of the prisms make any : for in a vessel made of polished plates of glass cemented together in the shape of a prism, and filled with water, there is the like success of the experiment according to the quantity of the refraction." But in the latter state- ment, as I shall afterwards remark (p. 432), Newton assumed an indifference which does not exist, and fell into an unfortunate mistake. In the science of sound it is shown that the pitch of a sound depends solely upon the number of impulses in a second, and the material exciting those impulses is a matter of indifference. Whatever fluid, air or water, gas or liquid, be forced into the Siren, the sound produced is the same ; and the material of which an organ-pipe is constructed does not at all affect the pitch of its sound. In the science of statical electricity it is an important principle that the nature of the interior of a conducting body is a matter of no importance. The electrical charge is confined to the conducting surface, and the interior remains in a neutral state. A hollow copper sphere takes exactly the same charge as a solid sphere of the same metal. Some of Faraday's most elegant and successful researches were devoted to the exclusion of conditions which previous experimentei-s had thought essential for the production of electrical phenomena. Davy asserted that no known fluids, except such as contain water, could be made the medium of connexion between the poles of a battery ; and some chemists believed that water was an essential agent in electro-chemical decomposition. Faraday gave abundant experiments to show that other fluids allowed of elec- trolysis, and he attributed the erroneous opinion to the very general use of water as a solvent, and its presence in most natural bodies.* It was, in fact, upon the weakest kind of negative evidence that the opinion had been founded. Many experimenters attributed peculiar powers to the poles of a battery, likening them to magnets, which, by their attractive powers, tear apart the elements of a sub- stance. By a beautiful series of experiments,* Faraday proved conclusively that, on the contrary, the substance of * .fixpenmentoZ Researches in Electrieity, vol. i. pp. 133, 134. * Ibid, vol L pp. 127, 162, &c. t. ! 4ft THE PRINCIPLES OP SCIENCE. [ClfAF. M O the poles is of no importance, being merely the path through which the electric force reaches the liquid acted upon. Poles of water, charcoal, and many diverse sub- stances, even air itself, produced similar results; if the chemical nature of the pole entered at all into the question, it was as a disturbing agent. It is an essential part of the theory of gravitation that the proximity of other attracting particles is without effect upon the attraction existing between any two molecules. Two pound weights weigh as much together as they do separately. Every pair of molecules in the world have, as it were, a private communication, apart from their rela- tions to all other molecules. Another undoubted result of experience pointed out by Newton ^ is that the weight of a body does not in the least depend upon its form or texture. It may be added that the temperature, electric condition, pressure, state of motion, chemical qualities, and all other circumstances concerning matter, except its mass, are indifferent as regards its gravitating power. As natural science progresses, physicists gain a kind of insight and tact in judging what qualities of a substance are likely to be concerned in any class of phenomena. The physical astronomer treats matter in one point of view, the chemist in another, and the students of physical optics, sound, mechanics, electricity, &c., make a fair division of the qualities among them. But errors will arise if too much confidence be placed in this independence of various kinds of phenomena, so that it is desirable from time to time, especially when any unexplained discrepancies come into notice, to question the indifference which is assumed to exist, and to test its real existence by appropriate experiments. SimpUJiccUion of Experiments, One of the most requisite precautions in experimentation is to vary only one circumstance at a time, and to main- tain all other cii'cumstances rigidly unchanged. There are two distinct reasons for this rule, the first and most ob- vious being that if we vary two conditions at a time, and ' Frineipiaf bk. iii. Prop. vi. Corollary I I I IX.] EXPERIMENT. 4^ find some effect, we cannot tell whether the effect is due to one or the other condition, or to both jointly. A second reason is that if no effect ensues we cannot safely conclude that either of them is indifferent ; for the one may have neutralised the effect of the other. In our symbolic logic AB -I- Ab was shown to be identical with A (p. 97), so that B denotes a circumstance which is indifferently present or absent. But if B always go together with another antecedent C, we cannot show the same inde- pendence, for ABC -I- Abe is not identical with A and none of our logical processes enables us to reduce it to A. If we want to prove that oxygen is necessary to life, we must not put a rabbit into a vessel from which the oxygen has been exhausted by a burning candle. We should then have not only an absence of oxygen, but an addition of carbonic acid, which may have been the destructive agent. For a similar reason Lavoisier avoided the use of atmo- spheric air in experiments on combustion, because air was not a simple substance, and the presence of nitrogen might impede or even alter the effect of oxygen. As Lavoisier remarks,* " In performing experiments, it is a necessary principle, which ought never to be deviated from, that they be simplified as much as possible, and that every circumstance capable of rendering their results complicated be carefully removed." It has also been well said by Cuvier ' that the method of physical inquiry consists in isolating bodies, reducing them to their utmost simplicity, and bringing each of their properties separately into action, either mentally or by experiment. The electro-magnet has been of the utmost service in the investigation of the magnetic properties of matter, by allowing of the production or removal of a most powerful magnetic force without disturbing any of the other ar- rangements of the experiment. Many of Faraday's most valuable experiments would have been impossible had it been necessary to introduce a heavy permanent magnet, which could not be suddenly moved without shaking the whole apparatus, disturbing the air, producing currents by changes of temperature, &c. The electro-magnet is » Layoisier's CJumittry, translated by Kerr, p. 103. • Cuvier's Animal Kingdom^ introduction. Dp i. 2. 424 THE PRINCIPLES OF SCIENCB. [chat. xu.] EXPERIMENT. 425 m • perfectly under control, and its influence can be brought into action, reversed, or stopped by merely touching a button. Thus Faraday was enabled to prove the rotation of the plane of circularly polarised light by the fact that certain light ceased to be visible when the electric current of the magnet was cut off, and re-appeared when the current was made. " These phenomena," he says, " could be reversed at pleasure, and at any instant of time, and upon any occasion, showing a perfect dependence of cause and effect." ^ It was Newton's omission to obtain the solar spectrum under the simplest conditions which prevented him from discovering the dark lines. Using a broad beam of light which had passed through a round hole or a triangular slit, he obtained a brilliant spectrum, but one in which many different coloured rays overlapped each other. In the recent history of the science of the spectrum, one main difficulty has consisted in the mixture of the lines of several different substances, which are usually to be found in the light of any flame or spark. It is seldom possible to obtain the light of any element in a perfectly simple manner. Angstrom greatly advanced this branch of science by examining the light of the electric spark when formed between poles of various metals, and in the presence of various gases. By varying the pole alone, or the gaseous medium alone, he was able to discriminate correctly be- tween the lines due to the metal and those due to the surrounding gas.^ Failure in the Simplification of Experiments, In some cases it seems to be impossible to carry out the rule of varying one circumstance at a time. When we attempt to obtain two instances or two forms of experi- ment in which a single circumstance shall be present in one case and absent in another, it may be found that this single circumstance entails others. Benjamin Franklin's experiment concerning the comparative absorbing powers of different colours is well known. " I took/* he says, " a ' Experimental Researches in Electricityf vol. iiL p. 4. * Philosophical Magazine, 4th Series, vol. ix. p. 327. number of little square pieces of broadcloth from a tailor's pattern card, of various colours. They were black, deep blue, lighter blue, green, purple, red, yellow, white, and other colours and shades of colour. I laid them all out upon the snow on a bright sunshiny morning. In a few hours the black, being most warmed by the sun, was sunk 80 low as to be below the stroke of the sun's rays ; the dark blue was almost as low ; the lighter blue not quite so much as the dark ; the other colours less as they were lighter. The white remained on the surface of the snow, not having entered it at all." This is a very elegant and apparently simple experiment ; but when Leslie had com- pleted liis series of researches upon the nature of heat, he came to the conclusion that the colour of a surface has very little effect upon the radiating power, the mechanical nature of the surface appearing to be more influentiaL He remarks ^ that " the question is incapable of being posi- tively resolved, since no substance can be made to assume difTerent colours without at the same time changing its internal structure." Recent investigation has shown that the subject is one of considerable complication, because the absorptive power of a surface may be different accord- ing to the character of the rays which fall upon it ; but there can be no doubt as to the acuteness with which Leslie points out the difficulty. In Well's investigations concerning the nature of dew, we have, again, very complicated conditions. If we expose plates of various material, such as rough iron, glass, polished metal, to the midnight sky, they will be dewed in various degrees j but since these plates differ both in the nature of the surface and the conducting power of the material, it would not be plain whether one or both circumstances were of importance. We avoid this difficulty by exposing the same material polished or varnished, so as to present dif- ferent conditions of surface ; * and again by exposing different substances with the same kind of surface. When we are quite unable to isolate circumstances we must resort to the procedure described by Mill under the name of the Joint Method of Agreement and Difference * Inquiry into tJie Naiure of Heai, p. 95. * Herschel, Preliminary Dtscoursey p. 161. 6 Ji / i tit 1 f i If 1 ' I 4S6 THE PRINCIPLES OF SCIENCE. [oHAr. We must collect as many instances as possible in which a given circumstance produces a given result, and as many as possible in which the absence of the circumstance is / followed by the absence of the result. To adduce his ^ example, we cannot experiment upon the cause of double refraction in Iceland spar, because we cannot alter its irystalline condition without altering it altogether, nor can we find substances exactly like calc spar in every circum- stance except one. We resort therefore to the method of comparing together all known substances which have the property of doubly-refracting light, and we find that they agree in being crystalline.^ This indeed is nothing but an ordinary process of perfect or probable induction, already partially described, and to be further discussed under Classification. It may be added that the subject does admit of perfect experimental treatment, since glass, when compressed in one direction, becomes capable of doubly- refracting light, and as there is probably no alteration in the glass but change of elasticity, we learn that the power of double refraction is probably due to a difference of elasticity in different directions. BenumU of UsiieU Conditions, One of the great objects of experiment is to enable us to judge of the behaviour of substances under conditions widely different from those which prevail upon the surface of the earth. We live in an atmosphere which does not vary beyond certain narrow limite in temperature or pressure. Many of the powers of nature, such as gravity, which constantly act upon us, are of almost fixed amount. Now it will afterwards be shown that we cannot apply a quantitative law to circumstances nmch differing from those in which it was observed. In the other planets, the sun, the stars, or remote parts of the Universe, the con- ditions of existence must often be widely different from what we commonly experience hera Hence our know- ledge of nature must remain restricted and hypothetical, unless we can subject substances to unusual conditions by suitable experiments. » Sydem of Logic, bk. iii. chap. viii. § 4, 5th ed, toI. i. p. 433. XIX.] BXPERIMENT. 427 The electric arc is an invaluable means of exposing metals or other conducting substances to the highest known temperatui*e. By its aid we learn not only that all the metals can be vaporised, but that they all give off distinctive i-ays of light At the other extremity of the scale, the intensely powerful freezing mixture devised by Faraday, consisting of solid carbonic acid and ether mixed in vacuo, enables us to observe the nature of substances at temperatures immensely below any we meet with naturally on the earth's surface. We can hardly realise now the importance of the in- vention of the air-pump, previous to which invention it was exceedingly difficult to experiment except under the ordinary pressure of the atmosphere. The Torricellian vacuum had been employed by the philosophers of the Accademia del Cimento to show the behaviour of water, smoke, sound, magnets, electric substances, &c., in vacuo, but their experiments were often unsuccessful from the difficulty of excluding air.^ Among the most constaitt circumstances under which we live is the force of gravity, which does not vary, except by a slight fraction of its amount, in any part of the earth's crust or atmosphere to which we can attain. This force is sufficient to overbear and disguise various actions, for in- stance, the mutual gravitation of small bodies. It was an interesting experiment of Plateau to neutralise the action of gravity by placing substances in liquids of exactly the same specific gravity. Thus a quantity of oil poured into the middle of a suitable mixture of alcohol and water assumes a spherical shape; on being made to rotate it becomes spheroidal, and then successively separates into a ring and a group of spherules. Thus we have an illustration of the mode in which the, planetary system may have been produced,* though the extreme difference of scale prevents our arguing with confidence from the experiment to the conditions of the nebidar theory. It is possible that the so-called elements are elementary only to us, because we are restricted to temperatures at which they are fixed. Lavoisier carefully defined an ' Jit$ttyei of Nalwral ^aperimnUs nubde in th$ Accademia del Cimento. Englished by Richard Waller, 1684, p. 40, &c - Plateau, Taylor's Sc%«iU\/ie Memoirt, toL ir. pp. 16—43. 4] !! ill t. A Vf I^^B f 1 m ^^H, 1 ; 91 »l \-l h'M ioi 4S8 THE PRINCIPLES OP SCIENCE. [CUAP. element as a substance which cannot be decomposed by any knovm means ; but it seems almost certain that some series of elements, for instance Iodine, Bromine, and Chlo- rine, are really compounds of a simpler substance. We must look to the production of intensely high temperatures, yet quite beyond our means, for the decomposition of these so-called elements. Possibly in this age and part of the universe the dissipation of energy has so far proceeded that there are no sources of heat sufficiently intense to effect the decomposition. Interference of Unsuspected Conditions. It may happen that we are not aware of all the conditions under which our researches are made. Some substance ^j// may be present or some power may be in action, which escapes the most vigilant examination. Not being awai-e of its existence, we are unable to take proper measures to exclude it, and thus determine the share which it has in the results of our experiment^. There can be no doubt that the alchemists were misled and encouraged in their vain attempts by the unsuspected presence of traces of gold and silver in the substances they proposed to trans- mute. Lead, as drawn from the smelting furnace, almost always contains some silver, and gold is associated with many other metals. Thus small quantities of noble metal would often appear as the result of experiment and raise delusive hopes. In more than one case the unsuspected presence of common salt in the air has caused great trouble. In the early experiments on electrolysis it was found that when water was decomposed, an acid and an alkali were produced at the poles, together with oxygen and hydrogen. In the absence of" any other explanation, some chemists rushed to the conclusion that electricity must have the power of generating acids and alkalies, and one chemist thought he had discovered a new substance called electric acid. But Davy proceeded to a systematic investigation of the circumstances, by varying the conditions. Changing the glass vessel for one of agate or gold, he found that far less alkali was produced ; excluding impurities by the use of carefully distilled water, he found that the quantities of ^ N ZIZ.] EXPERIMENT. 429 acid and alkali were still further diminished ; and having thus obtained a clue to the cause, he completed the ex- clusion of impurities by avoiding contact with his fingers, and by placing the apparatus under an exhausted receiver, no acid or alkali being then detected. It would be difficult to meet with a more elegant case of the detection of a condition previously unsuspected.^ It is remarkable that the presence of common salt in the air, proved to exist by Davy, nevertheless continued a stumbling-block in the science of spectrum analysis, and probably prevented men, such as Brewster, Herschel, and Talbot, from anticipating by thirty years the discoveries of Bunsen and Kirchhoff. As I pointed out,* the utility of the spectrum was known in the middle of the last century to Thomas Melvill, a talented Scotch physicist, who died at the early age of 27 years.^ But Melvill was struck in his examination of coloured flames by the extraordinary predominance of homogeneous yellow light, which was due to some circumstance escaping his atten- tion. Wollaston and Fraunhofer were equally struck by the prominence of the yellow line in the spectrum of nearly every kind of light. Talbot expressly recommended tlie use of the prism for detecting the presence of substances by what we now call spectrum analysis, but he found that all substances, however different the light they yielded in other respects, were identical as regards the production of yellow light. Talbot knew that the salts of soda gave this coloured light, but in spite of Davy's previous difficulties with salt in electrolysis, it did not occur to him to assert that where the light is, there sodium must be. He sug- gested water as the most likely source of the yellow light, because of its frequent presence, but even substances which were apparently devoid of water gave the same yellow light.* Brewster and Herschel both experimented * Philosophical Transactions [1826], vol. cxvi. pp. 388, 389. Works of Sir Humphry Davy, vol. v. pp. i — 12. * National lieview^ July, 1801, p. 13. * His published works are contained in The Edinburgh Physical and Literary Essays, vol. ii. p. 34 ; Philosophical Transactions [1753], vol xlviii. p. 261 ; see also Morgan's Papers in Philosophical Trans- ft^tiiont [1755], vol. IxiT. p. 190. Itwmrgh I ill Edit Journal of Science, voL v. p. 79. 430 f THE PRINCIPLES OP SCIENCE. [CBAF. ♦ . If, upoa flames almost at the same time as Talbot, and Herschel unequivocally enounced the principle of spec- trum analysis.^ Nevertheless Brewster, after numerous experiments attended with great trouble and disappoint- ment, found that yellow light might be obtained from the combustion of almost any substance. It was not until 1856 that Swan discovered that an almost infinitesimal quantity of sodium chloride, say a millionth part of a grain, was sufficient to tinge a flame of a bright yellow colour. The universal diffusion of the salts of sodium, joined to this unique light-producing power, was thus shown to be the unsuspected condition which had destroyed the confi- dence of all previous experimenters in the use of the prism. Some references concerning the history of this curious point are given below.* In the science of radiant heat, early inquirers were led to the conclusion that radiation proceeded only from the surface of a solid, or from a very small depth below it But they happened to experiment upon surfaces covered by coats of varnish, which is highly athermanous or opaque to heat Had they properly varied the character of the surface, using a highly diathermanous substance like rock salt, they would have obtained very different results.' One of the most extraordinary instances of an erroneous opinion due to overlooking interfering agents is that con- cerning the increase of rainfall near to the earth's surface. More than a century ago it was observed that rain-gauges placed upon church steeples, house tops, and other elevated places, gave considerably less rain than if they were on the ground, and it has been recently shown that the variation is most rapid in the close neighbourhood of the ground.* All kinds of theories have been started to explain this phenomenon ; but I have shown ' that it is simply due to * Eneyclop<Bdia Mdropolitanaf art LigJU, § 524; Herschcri F(Mnxliar Lectures^ p. 266. ■ Talbot, Philasophienl Magaziney vd Series, vol. ix. p. i (1836); Brewster, Transaetiotu of tJu Royal Society 0/ Edinburgh [1823I vol. ii. pp.433, 455 ; Swan, ibid. [1856] vol xxL p. 41 1 ; rhilotophical Magazine, 4th Series, vol xx. p. 173 [Sept 1860J ; Boecoe, i^^keUntm Analytiif Lecture IIL * Balfour Stewart, EUmentary Treatiie on Ueai^ p. 192. * British Association, Liverpool, 1870. Rewtri on Kaii^faU, p. I7d * PhUoiophieal Magaxiue. Dec. 1861. 4th Series, vol. xxii p. 421^ i Ii xiz.] EXPERIMENT. 431 the interference of wind, which deflects more or less rain from all the gauges which are exposed to it. The great magnetic power of iron renders it a source of disturbance in magnetic experiments. In building a mag- netic observatory great care must therefore be taken that no iron is employed in the construction, and that no masses of iron are near at hand. In some cases magnetic observations have been seriously disturbed by the existence of masses of iron ore in the neighbourhood. In Faraday's experiments upon feebly magnetic or diamagnetic substances he took the greatest precautions against the presence of disturbing substances in the copper wire, wax, paper, and other articles used in suspending the test objects. It was his custom to try the effect of the magnet upon the appa- ratus in the absence of the object of experiment, and with- out this preliminary trial no confidence could be placed in the results.^ Tyndall has also employed the same mode for testing the freedom of electro-magnetic coils from iron, and was thus enabled to obtain them devoid of any cause of disturbance.* It is worthy of notice that in the very infancy of the science of magnetism, the acute experimen- talist Gilbert correctly accounted for the opinion existing in his day that magnets would attract silver, by pointing out that the silver contained iron. Even when we are not aware by previous experience of the probable presence of a special disturbing agent, we ought not to assume the absence of unsuspected inter- ference. If an experiment is of really high importance, so that any considerable branch of science rests upon it, we ought to try it again and again, in as varied conditions as possibla We should intentionally disturb the apparatus in various ways, so as if possible to hit by accident upon any weak point Especially when our results are more regular than we have fair grounds for anticipating, ought we to suspect some peculiarity in the apparatus which causes it to measure some other phenomenon than that in question, just as Foucault's pendulum almost always in- dicates the movement of the axes of its own elliptic path instead of the rotation of the globe. * Experimental Researches in Electricity , vol. iil p. 84. &c. * Lectures on Heal, p. 21. 5 ! I ' i\ 432 THE PRINCIPLES OF SCIENCR [OHAP. n-. It was in this cautious spirit that Baily acted in his experiments on the density of the earth. The accuracy of his results depended upon the elimination of all disturb- ing influences, so that the oscillation of his torsion balance should measure gravity alone. Hence he varied the appa- ratus in many ways, changing the small balls subject to attraction, changing the connecting rod, and the means of suspension. He observed the effect of disturbances, such as the presence of visitors, the occurrence of violent storms, &c., and as no real alteration was produced in the results, he confidently attributed them to gravity.* Newton would probably have discovered the mode of constructing achromatic lenses, but for the unsuspected effect of some sugar of lead which he is supposed to have dissolved in the water of a prism. He tried, by means of a glass prism combined with a water prism, to produce dispersion of light without refraction, and if he had succeeded there would liave been an obvious mode of producing refraction without dispersion. His failure is attributed to his adding lead acetate to the water for the purpose of increasing its refractive power, the lead having a high dispersive power which frustrated his purpose.' Judging from Newton's remarks, in the Philosophical Transactions, it would appear as if he had not, without many unsuccessful trials, despaired of the construction of achromatic glasses.' The Academicians of Cimento, in their early and in- genious experiments upon the vacuum, were often misled by the mechanical imperfections of their ap})aratus. They concluded that the air had nothing to do with the produc- tion of sounds, evidently because their vacuum was not sufficiently perfect. Otto von Guericke fell into a like mistake in the use of his newly-constructed air-pump, doubtless from the unsuspected presence of air sufficiently dense to convey the sound of the bell. It is hardly requisite to point out that the doctrine of spontaneous generation is due to the unsuspected presence » Baily, Memoin of tJie Royal Aiironamieal Society, vol. xir. pn. 29, 30. '' *^*^ *- Grant, History of Phytical Aitronomy, p. 531. • Fkilotophical Transactiont, abridged by Lowthorp, 4th edition, Tol i. p. 20a. f^\ XIX.] EXPERIMENT. 433 of germs, even after the most careful efforts to exclude them, and in the case of many diseases, both of animals and plants, germs which we have no means as yet of de- tecting are doubtless the active cause. It has long been II subject of dispute, again, whether the plants which spring from newly turned land grow from seeds long buried in that land, or from seeds brought by the wind. Argument is unphilosophical when direct trial can readily be applied ; for by turning up some old ground, and covering a portion of it with a glass case, the conveyance of seeds by the wind can be entirely prevented, and if the same plants appear within and without the case, it will become clear that the seeds are in the earth. By gross oversight some experimenters have thought before now that crops of ry( nad sprung up where oats had been sown. Blind or Test Experiments, Every conclusive experiment necessarily consists in the comparison of results between two different combinations of circumstances. To give a fair probability that A is the cause of X, we must maintain invariable all surrounding objects and conditions, and we must then show that where A is X is, and where A is not X is not. This cannot really be accomplished in a single trial. If, for instance, a chemist places a certain suspected substance in Marsh's test apparatus, and finds that it gives a small deposit of metallic arsenic, he cannot be sure that the arsenic really proceeds from the suspected substance ; the impurity of the zinc or sulphuric acid may have been the cause"^ of its appearance. It is therefore the practice of chemists to make what they call a blind experiment, that is to try whether arsenic appears in the absence of the suspected substance. The same precaution ought to be taken in all important analytical operations. Indeed, it is not merely a precaution, it is an essential part of any experiment. If the blind trial be not made, the chemist merely assumes that he knows what would happen. Whenever we assert that because A and X are found together A is the cause of X. we assume that if A were absent X would be absent. But wherever it is possible, we ought not to take this as a mere assumption, or even as a matter of inference. F ? n \\ i! 434 TUB PRINCIPLES OP SCIENCK. [chap tix.] EXPERIMENT. 43ft Experience is ultimately the basis of all our inferences, but if we can bring immediate experience to bear upon the point in question we should not trust to anything more remote and liable to error. When Faraday examined the magnetic properties ofthe bearing apparatus, in the absence of the substance to be experimented on, he really made a blind experiment (p. 431). We ought, also, to test the accuracy of a method of ex- ^ perim